%------------------------------------------------------------------------------
% File : DT2H2X---1.9.5
% Problem : ITP056^1 : TPTP v9.2.1. Released v7.5.0.
% Transfm : none
% Format : tptp:raw
% Command : /export/starexec/sandbox/solver/bin/run_DT2H2X /export/starexec/sandbox/benchmark/theBenchmark.p 300
% Computer : n019.cluster.edu
% Model : x86_64 x86_64
% CPU : Intel(R) Xeon(R) CPU E5-2620 v4 2.10GHz
% Memory : 8042.1875MB
% OS : Linux 3.10.0-693.el7.x86_64
% CPULimit : 300s
% WCLimit : 300s
% DateTime : Wed Feb 25 08:55:11 AM UTC 2026
% Result : Theorem 0.30s 1.86s
% Output : Refutation 0.30s
% Verified :
% SZS Type : Refutation
% Derivation depth : 6
% Number of leaves : 3
% Syntax : Number of formulae : 12 ( 8 unt; 0 typ; 0 def)
% Number of atoms : 16 ( 15 equ; 0 cnn)
% Maximal formula atoms : 2 ( 1 avg)
% Number of connectives : 73 ( 7 ~; 3 |; 0 &; 62 @)
% ( 0 <=>; 1 =>; 0 <=; 0 <~>)
% Maximal formula depth : 4 ( 2 avg)
% Number of types : 4 ( 3 usr)
% Number of type conns : 0 ( 0 >; 0 *; 0 +; 0 <<)
% Number of symbols : 90 ( 88 usr; 5 con; 0-4 aty)
% Number of variables : 3 ( 0 ^; 3 !; 0 ?; 3 :)
% Comments :
%------------------------------------------------------------------------------
thf(type_def_5,type,
nat: $tType ).
thf(type_def_7,type,
list_c1059388851t_unit: $tType ).
thf(type_def_8,type,
config256849571t_unit: $tType ).
thf(func_def_0,type,
list_c1059388851t_unit: $tType ).
thf(func_def_1,type,
config256849571t_unit: $tType ).
thf(func_def_2,type,
nat: $tType ).
thf(func_def_3,type,
minus_minus_nat: nat > nat > nat ).
thf(func_def_4,type,
one_one_nat: nat ).
thf(func_def_5,type,
plus_plus_nat: nat > nat > nat ).
thf(func_def_6,type,
prefix1615116500t_unit: list_c1059388851t_unit > list_c1059388851t_unit > $o ).
thf(func_def_7,type,
last_c571238084t_unit: list_c1059388851t_unit > config256849571t_unit ).
thf(func_def_8,type,
nil_co1338500125t_unit: list_c1059388851t_unit ).
thf(func_def_9,type,
nth_co1649820636t_unit: list_c1059388851t_unit > nat > config256849571t_unit ).
thf(func_def_10,type,
size_s1406904903t_unit: list_c1059388851t_unit > nat ).
thf(func_def_11,type,
ord_less_nat: nat > nat > $o ).
thf(func_def_12,type,
ord_less_eq_nat: nat > nat > $o ).
thf(func_def_13,type,
fe: nat > list_c1059388851t_unit ).
thf(func_def_14,type,
n0: nat ).
thf(func_def_15,type,
n: nat ).
thf(func_def_30,type,
sP0: ( nat > $o ) > ( nat > $o ) > ( nat > $o ) > ( nat > $o ) > $o ).
thf(func_def_31,type,
sP1: ( nat > $o ) > ( nat > $o ) > ( nat > $o ) > ( nat > $o ) > $o ).
thf(func_def_32,type,
sP2: ( nat > $o ) > ( nat > $o ) > ( nat > $o ) > ( nat > $o ) > $o ).
thf(func_def_33,type,
sP3: nat > nat > nat > $o ).
thf(func_def_34,type,
sP4: nat > nat > nat > $o ).
thf(func_def_35,type,
sP5: nat > nat > nat > $o ).
thf(func_def_36,type,
sP6: ( nat > $o ) > ( nat > $o ) > ( nat > $o ) > ( nat > $o ) > $o ).
thf(func_def_37,type,
sK7: ( nat > nat > $o ) > nat ).
thf(func_def_38,type,
sK8: ( nat > nat > $o ) > nat ).
thf(func_def_39,type,
sK9: ( nat > nat > $o ) > nat ).
thf(func_def_40,type,
sK10: ( nat > nat > $o ) > nat ).
thf(func_def_41,type,
sK11: ( nat > nat > $o ) > nat ).
thf(func_def_42,type,
sK12: ( nat > nat > $o ) > nat ).
thf(func_def_43,type,
sK13: ( nat > nat > $o ) > nat ).
thf(func_def_44,type,
sK14: ( nat > nat > $o ) > nat ).
thf(func_def_45,type,
sK15: ( nat > nat > $o ) > nat ).
thf(func_def_46,type,
sK16: list_c1059388851t_unit > list_c1059388851t_unit > nat ).
thf(func_def_47,type,
sK17: nat > nat > nat ).
thf(func_def_48,type,
sK18: ( nat > nat ) > nat ).
thf(func_def_49,type,
sK19: ( nat > nat ) > nat ).
thf(func_def_50,type,
sK20: ( nat > nat ) > nat ).
thf(func_def_51,type,
sK21: ( nat > nat ) > nat ).
thf(func_def_52,type,
sK22: nat > nat ).
thf(func_def_53,type,
sK23: nat > nat ).
thf(func_def_54,type,
sK24: nat > nat ).
thf(func_def_55,type,
sK25: ( nat > $o ) > ( nat > $o ) > ( nat > $o ) > ( nat > $o ) > nat ).
thf(func_def_56,type,
sK26: nat > ( nat > $o ) > ( nat > $o ) > nat ).
thf(func_def_57,type,
sK27: nat > ( nat > $o ) > ( nat > $o ) > nat ).
thf(func_def_58,type,
sK28: ( nat > nat ) > nat ).
thf(func_def_59,type,
sK29: ( nat > nat ) > nat ).
thf(func_def_60,type,
sK30: ( nat > nat ) > nat ).
thf(func_def_61,type,
sK31: ( nat > nat ) > nat ).
thf(func_def_62,type,
sK32: ( nat > $o ) > ( nat > $o ) > ( nat > $o ) > ( nat > $o ) > nat ).
thf(func_def_63,type,
sK33: nat > ( nat > $o ) > ( nat > $o ) > nat ).
thf(func_def_64,type,
sK34: nat > ( nat > $o ) > ( nat > $o ) > nat ).
thf(func_def_65,type,
sK35: ( nat > $o ) > ( nat > $o ) > ( nat > $o ) > ( nat > $o ) > nat ).
thf(func_def_66,type,
sK36: nat > ( nat > $o ) > ( nat > $o ) > nat ).
thf(func_def_67,type,
sK37: nat > ( nat > $o ) > ( nat > $o ) > nat ).
thf(func_def_68,type,
sK38: nat > list_c1059388851t_unit ).
thf(func_def_69,type,
sK39: nat > nat ).
thf(func_def_70,type,
sK40: ( nat > $o ) > nat ).
thf(func_def_71,type,
sK41: ( nat > $o ) > nat > nat ).
thf(func_def_72,type,
sK42: ( nat > $o ) > nat > nat ).
thf(func_def_73,type,
sK43: nat > ( nat > config256849571t_unit > $o ) > list_c1059388851t_unit ).
thf(func_def_74,type,
sK44: nat > ( nat > config256849571t_unit > $o ) > nat ).
thf(func_def_75,type,
sK45: nat > ( nat > config256849571t_unit > $o ) > config256849571t_unit ).
thf(func_def_76,type,
sK46: list_c1059388851t_unit > nat > ( nat > config256849571t_unit > $o ) > nat ).
thf(func_def_77,type,
sK47: nat > nat ).
thf(func_def_78,type,
sK48: ( nat > nat ) > nat ).
thf(func_def_79,type,
sK49: ( nat > nat ) > nat ).
thf(func_def_80,type,
sK50: nat > nat ).
thf(func_def_81,type,
sK51: ( nat > nat ) > nat ).
thf(func_def_82,type,
sK52: ( nat > nat ) > nat ).
thf(func_def_83,type,
sK53: nat > nat > nat ).
thf(func_def_84,type,
sK54: ( list_c1059388851t_unit > $o ) > list_c1059388851t_unit ).
thf(func_def_85,type,
sK55: ( nat > nat ) > nat ).
thf(func_def_86,type,
sK56: ( nat > nat ) > nat ).
thf(func_def_87,type,
sK57: nat > nat ).
thf(func_def_88,type,
sK58: nat > nat ).
thf(func_def_89,type,
sK59: ( nat > nat ) > nat ).
thf(func_def_90,type,
sK60: ( nat > nat ) > nat ).
thf(func_def_91,type,
sK61: nat > nat ).
thf(func_def_92,type,
sK62: ( nat > nat ) > nat ).
thf(func_def_93,type,
sK63: ( nat > nat ) > nat ).
thf(func_def_94,type,
sK64: nat > nat ).
thf(func_def_95,type,
sK65: ( nat > $o ) > nat ).
thf(func_def_96,type,
sK66: ( nat > $o ) > nat > nat ).
thf(func_def_97,type,
sK67: ( nat > $o ) > nat ).
thf(func_def_98,type,
sK68: ( nat > nat ) > nat ).
thf(func_def_99,type,
sK69: ( nat > nat ) > nat ).
thf(func_def_100,type,
sK70: ( nat > $o ) > nat ).
thf(func_def_101,type,
sK71: ( nat > $o ) > ( nat > $o ) > ( nat > $o ) > ( nat > $o ) > nat ).
thf(func_def_102,type,
sK72: nat > ( nat > $o ) > ( nat > $o ) > nat ).
thf(func_def_103,type,
sK73: nat > ( nat > $o ) > ( nat > $o ) > nat ).
thf(func_def_105,type,
ph75:
!>[X0: $tType] : X0 ).
thf(f1345,plain,
$false,
inference(subsumption_resolution,[status(thm)],[f1344,f1182]) ).
thf(f1182,plain,
( ( fe @ n )
!= nil_co1338500125t_unit ),
inference(cnf_transformation,[status(thm)],[f1]) ).
thf(f1,axiom,
( ( fe @ n )
!= nil_co1338500125t_unit ),
file('/export/starexec/sandbox/tmp/tmp.iTgpky4eW3/DTF2THF_32211.p',fact_0_FeNNotEmpty) ).
thf(f1344,plain,
( ( fe @ n )
= nil_co1338500125t_unit ),
inference(trivial_inequality_removal,[status(thm)],[f1342]) ).
thf(f1342,plain,
( ( ( fe @ n )
= nil_co1338500125t_unit )
| ( ( last_c571238084t_unit @ ( fe @ n ) )
!= ( last_c571238084t_unit @ ( fe @ n ) ) ) ),
inference(superposition,[status(thm)],[f1001,f1029]) ).
thf(f1029,plain,
! [X0: list_c1059388851t_unit] :
( ( ( last_c571238084t_unit @ X0 )
= ( nth_co1649820636t_unit @ X0 @ ( minus_minus_nat @ ( size_s1406904903t_unit @ X0 ) @ one_one_nat ) ) )
| ( nil_co1338500125t_unit = X0 ) ),
inference(cnf_transformation,[status(thm)],[f655]) ).
thf(f655,plain,
! [X0: list_c1059388851t_unit] :
( ( nil_co1338500125t_unit = X0 )
| ( ( last_c571238084t_unit @ X0 )
= ( nth_co1649820636t_unit @ X0 @ ( minus_minus_nat @ ( size_s1406904903t_unit @ X0 ) @ one_one_nat ) ) ) ),
inference(ennf_transformation,[status(thm)],[f2]) ).
thf(f2,axiom,
! [X0: list_c1059388851t_unit] :
( ( nil_co1338500125t_unit != X0 )
=> ( ( last_c571238084t_unit @ X0 )
= ( nth_co1649820636t_unit @ X0 @ ( minus_minus_nat @ ( size_s1406904903t_unit @ X0 ) @ one_one_nat ) ) ) ),
file('/export/starexec/sandbox/tmp/tmp.iTgpky4eW3/DTF2THF_32211.p',fact_1_LastIsLastIndex) ).
thf(f1001,plain,
( ( last_c571238084t_unit @ ( fe @ n ) )
!= ( nth_co1649820636t_unit @ ( fe @ n ) @ ( minus_minus_nat @ ( size_s1406904903t_unit @ ( fe @ n ) ) @ one_one_nat ) ) ),
inference(cnf_transformation,[status(thm)],[f574]) ).
thf(f574,plain,
( ( last_c571238084t_unit @ ( fe @ n ) )
!= ( nth_co1649820636t_unit @ ( fe @ n ) @ ( minus_minus_nat @ ( size_s1406904903t_unit @ ( fe @ n ) ) @ one_one_nat ) ) ),
inference(flattening,[status(thm)],[f216]) ).
thf(f216,negated_conjecture,
( ( last_c571238084t_unit @ ( fe @ n ) )
!= ( nth_co1649820636t_unit @ ( fe @ n ) @ ( minus_minus_nat @ ( size_s1406904903t_unit @ ( fe @ n ) ) @ one_one_nat ) ) ),
inference(negated_conjecture,[status(cth)],[f215]) ).
thf(f215,conjecture,
( ( last_c571238084t_unit @ ( fe @ n ) )
= ( nth_co1649820636t_unit @ ( fe @ n ) @ ( minus_minus_nat @ ( size_s1406904903t_unit @ ( fe @ n ) ) @ one_one_nat ) ) ),
file('/export/starexec/sandbox/tmp/tmp.iTgpky4eW3/DTF2THF_32211.p',conj_0) ).
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.12 % Problem : ITP056^1 : TPTP v9.2.1. Released v7.5.0.
% 0.12/0.13 % Command : /export/starexec/sandbox/solver/bin/run_DT2H2X /export/starexec/sandbox/benchmark/theBenchmark.p 300
% 0.16/0.34 % Computer : n019.cluster.edu
% 0.16/0.34 % Model : x86_64 x86_64
% 0.16/0.34 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.16/0.34 % Memory : 8042.1875MB
% 0.16/0.34 % OS : Linux 3.10.0-693.el7.x86_64
% 0.16/0.34 % CPULimit : 300
% 0.16/0.34 % WCLimit : 300
% 0.16/0.34 % DateTime : Tue Feb 24 08:24:54 EST 2026
% 0.16/0.34 % CPUTime :
% 0.16/0.34 Running /export/starexec/sandbox/solver/bin/run_DT2H2X /export/starexec/sandbox/benchmark/theBenchmark.p 300
% 0.19/0.46 ---- Original DTF file ---
% 0.19/0.46 thf(spec,logic,$$dhol).
% 0.19/0.46 %------------------------------------------------------------------------------
% 0.19/0.46 % File : ITP056^1 : TPTP v9.2.1. Released v7.5.0.
% 0.19/0.46 % Domain : Interactive Theorem Proving
% 0.19/0.46 % Problem : Sledgehammer FLPTheorem problem prob_1320__3306644_1
% 0.19/0.46 % Version : Especial.
% 0.19/0.46 % English :
% 0.19/0.46
% 0.19/0.46 % Refs : [BH+15] Blanchette et al. (2015), Mining the Archive of Formal
% 0.19/0.46 % : [Des21] Desharnais (2021), Email to Geoff Sutcliffe
% 0.19/0.46 % Source : [Des21]
% 0.19/0.46 % Names : FLPTheorem/prob_1320__3306644_1 [Des21]
% 0.19/0.46
% 0.19/0.46 % Status : Theorem
% 0.19/0.46 % Rating : 0.22 v9.1.0, 0.12 v9.0.0, 0.30 v8.2.0, 0.15 v8.1.0, 0.18 v7.5.0
% 0.19/0.46 % Syntax : Number of formulae : 231 ( 69 unt; 16 typ; 0 def)
% 0.19/0.46 % Number of atoms : 543 ( 171 equ; 0 cnn)
% 0.19/0.46 % Maximal formula atoms : 12 ( 2 avg)
% 0.19/0.46 % Number of connectives : 1758 ( 62 ~; 13 |; 22 &;1395 @)
% 0.19/0.46 % ( 0 <=>; 266 =>; 0 <=; 0 <~>)
% 0.19/0.46 % Maximal formula depth : 14 ( 7 avg)
% 0.19/0.46 % Number of types : 4 ( 3 usr)
% 0.19/0.46 % Number of type conns : 55 ( 55 >; 0 *; 0 +; 0 <<)
% 0.19/0.46 % Number of symbols : 14 ( 13 usr; 4 con; 0-2 aty)
% 0.19/0.46 % Number of variables : 647 ( 28 ^; 587 !; 32 ?; 647 :)
% 0.19/0.46 % SPC : TH0_THM_EQU_NAR
% 0.19/0.46
% 0.19/0.46 % Comments : This file was generated by Sledgehammer 2021-02-23 15:32:41.347
% 0.19/0.46 %------------------------------------------------------------------------------
% 0.19/0.46 % Could-be-implicit typings (3)
% 0.19/0.46 thf(ty_n_t__List__Olist_It__AsynchronousSystem__Oconfiguration__Oconfiguration____ext_Itf__p_Mtf__v_Mtf__s_Mt__Product____Type__Ounit_J_J,type,
% 0.19/0.46 list_c1059388851t_unit: $tType ).
% 0.19/0.46
% 0.19/0.46 thf(ty_n_t__AsynchronousSystem__Oconfiguration__Oconfiguration____ext_Itf__p_Mtf__v_Mtf__s_Mt__Product____Type__Ounit_J,type,
% 0.19/0.46 config256849571t_unit: $tType ).
% 0.19/0.46
% 0.19/0.46 thf(ty_n_t__Nat__Onat,type,
% 0.19/0.46 nat: $tType ).
% 0.19/0.46
% 0.19/0.46 % Explicit typings (13)
% 0.19/0.46 thf(sy_c_Groups_Ominus__class_Ominus_001t__Nat__Onat,type,
% 0.19/0.46 minus_minus_nat: nat > nat > nat ).
% 0.19/0.46
% 0.19/0.46 thf(sy_c_Groups_Oone__class_Oone_001t__Nat__Onat,type,
% 0.19/0.46 one_one_nat: nat ).
% 0.19/0.46
% 0.19/0.46 thf(sy_c_Groups_Oplus__class_Oplus_001t__Nat__Onat,type,
% 0.19/0.46 plus_plus_nat: nat > nat > nat ).
% 0.19/0.46
% 0.19/0.46 thf(sy_c_ListUtilities_OprefixList_001t__AsynchronousSystem__Oconfiguration__Oconfiguration____ext_Itf__p_Mtf__v_Mtf__s_Mt__Product____Type__Ounit_J,type,
% 0.19/0.46 prefix1615116500t_unit: list_c1059388851t_unit > list_c1059388851t_unit > $o ).
% 0.19/0.46
% 0.19/0.46 thf(sy_c_List_Olast_001t__AsynchronousSystem__Oconfiguration__Oconfiguration____ext_Itf__p_Mtf__v_Mtf__s_Mt__Product____Type__Ounit_J,type,
% 0.19/0.46 last_c571238084t_unit: list_c1059388851t_unit > config256849571t_unit ).
% 0.19/0.46
% 0.19/0.46 thf(sy_c_List_Olist_ONil_001t__AsynchronousSystem__Oconfiguration__Oconfiguration____ext_Itf__p_Mtf__v_Mtf__s_Mt__Product____Type__Ounit_J,type,
% 0.19/0.46 nil_co1338500125t_unit: list_c1059388851t_unit ).
% 0.19/0.46
% 0.19/0.46 thf(sy_c_List_Onth_001t__AsynchronousSystem__Oconfiguration__Oconfiguration____ext_Itf__p_Mtf__v_Mtf__s_Mt__Product____Type__Ounit_J,type,
% 0.19/0.46 nth_co1649820636t_unit: list_c1059388851t_unit > nat > config256849571t_unit ).
% 0.19/0.46
% 0.19/0.46 thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__AsynchronousSystem__Oconfiguration__Oconfiguration____ext_Itf__p_Mtf__v_Mtf__s_Mt__Product____Type__Ounit_J_J,type,
% 0.19/0.46 size_s1406904903t_unit: list_c1059388851t_unit > nat ).
% 0.19/0.46
% 0.19/0.46 thf(sy_c_Orderings_Oord__class_Oless_001t__Nat__Onat,type,
% 0.19/0.46 ord_less_nat: nat > nat > $o ).
% 0.19/0.46
% 0.19/0.46 thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Nat__Onat,type,
% 0.19/0.46 ord_less_eq_nat: nat > nat > $o ).
% 0.19/0.46
% 0.19/0.46 thf(sy_v_fe____,type,
% 0.19/0.46 fe: nat > list_c1059388851t_unit ).
% 0.19/0.46
% 0.19/0.46 thf(sy_v_n0____,type,
% 0.19/0.46 n0: nat ).
% 0.19/0.46
% 0.19/0.46 thf(sy_v_n____,type,
% 0.19/0.46 n: nat ).
% 0.19/0.46
% 0.19/0.46 % Relevant facts (214)
% 0.19/0.46 thf(fact_0_FeNNotEmpty,axiom,
% 0.19/0.46 ( ( fe @ n )
% 0.19/0.46 != nil_co1338500125t_unit ) ).
% 0.19/0.46
% 0.19/0.46 % FeNNotEmpty
% 0.19/0.46 thf(fact_1_LastIsLastIndex,axiom,
% 0.19/0.46 ! [L: list_c1059388851t_unit] :
% 0.19/0.46 ( ( L != nil_co1338500125t_unit )
% 0.19/0.46 => ( ( last_c571238084t_unit @ L )
% 0.19/0.46 = ( nth_co1649820636t_unit @ L @ ( minus_minus_nat @ ( size_s1406904903t_unit @ L ) @ one_one_nat ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % LastIsLastIndex
% 0.19/0.46 thf(fact_2_Fou2,axiom,
% 0.19/0.46 ord_less_nat @ ( minus_minus_nat @ ( size_s1406904903t_unit @ ( fe @ n ) ) @ one_one_nat ) @ ( size_s1406904903t_unit @ ( fe @ n ) ) ).
% 0.19/0.46
% 0.19/0.46 % Fou2
% 0.19/0.46 thf(fact_3_AssmNoDecided_I1_J,axiom,
% 0.19/0.46 ord_less_nat @ n0 @ ( size_s1406904903t_unit @ ( fe @ n ) ) ).
% 0.19/0.46
% 0.19/0.46 % AssmNoDecided(1)
% 0.19/0.46 thf(fact_4_last__conv__nth,axiom,
% 0.19/0.46 ! [Xs: list_c1059388851t_unit] :
% 0.19/0.46 ( ( Xs != nil_co1338500125t_unit )
% 0.19/0.46 => ( ( last_c571238084t_unit @ Xs )
% 0.19/0.46 = ( nth_co1649820636t_unit @ Xs @ ( minus_minus_nat @ ( size_s1406904903t_unit @ Xs ) @ one_one_nat ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % last_conv_nth
% 0.19/0.46 thf(fact_5_Fou,axiom,
% 0.19/0.46 ord_less_eq_nat @ n0 @ ( minus_minus_nat @ ( size_s1406904903t_unit @ ( fe @ n ) ) @ one_one_nat ) ).
% 0.19/0.46
% 0.19/0.46 % Fou
% 0.19/0.46 thf(fact_6_one__natural_Orsp,axiom,
% 0.19/0.46 one_one_nat = one_one_nat ).
% 0.19/0.46
% 0.19/0.46 % one_natural.rsp
% 0.19/0.46 thf(fact_7_diff__commute,axiom,
% 0.19/0.46 ! [I: nat,J: nat,K: nat] :
% 0.19/0.46 ( ( minus_minus_nat @ ( minus_minus_nat @ I @ J ) @ K )
% 0.19/0.46 = ( minus_minus_nat @ ( minus_minus_nat @ I @ K ) @ J ) ) ).
% 0.19/0.46
% 0.19/0.46 % diff_commute
% 0.19/0.46 thf(fact_8_Ex__list__of__length,axiom,
% 0.19/0.46 ! [N: nat] :
% 0.19/0.46 ? [Xs2: list_c1059388851t_unit] :
% 0.19/0.46 ( ( size_s1406904903t_unit @ Xs2 )
% 0.19/0.46 = N ) ).
% 0.19/0.46
% 0.19/0.46 % Ex_list_of_length
% 0.19/0.46 thf(fact_9_neq__if__length__neq,axiom,
% 0.19/0.46 ! [Xs: list_c1059388851t_unit,Ys: list_c1059388851t_unit] :
% 0.19/0.46 ( ( ( size_s1406904903t_unit @ Xs )
% 0.19/0.46 != ( size_s1406904903t_unit @ Ys ) )
% 0.19/0.46 => ( Xs != Ys ) ) ).
% 0.19/0.46
% 0.19/0.46 % neq_if_length_neq
% 0.19/0.46 thf(fact_10_size__neq__size__imp__neq,axiom,
% 0.19/0.46 ! [X: list_c1059388851t_unit,Y: list_c1059388851t_unit] :
% 0.19/0.46 ( ( ( size_s1406904903t_unit @ X )
% 0.19/0.46 != ( size_s1406904903t_unit @ Y ) )
% 0.19/0.46 => ( X != Y ) ) ).
% 0.19/0.46
% 0.19/0.46 % size_neq_size_imp_neq
% 0.19/0.46 thf(fact_11_diff__diff__cancel,axiom,
% 0.19/0.46 ! [I: nat,N: nat] :
% 0.19/0.46 ( ( ord_less_eq_nat @ I @ N )
% 0.19/0.46 => ( ( minus_minus_nat @ N @ ( minus_minus_nat @ N @ I ) )
% 0.19/0.46 = I ) ) ).
% 0.19/0.46
% 0.19/0.46 % diff_diff_cancel
% 0.19/0.46 thf(fact_12_le__refl,axiom,
% 0.19/0.46 ! [N: nat] : ( ord_less_eq_nat @ N @ N ) ).
% 0.19/0.46
% 0.19/0.46 % le_refl
% 0.19/0.46 thf(fact_13_le__trans,axiom,
% 0.19/0.46 ! [I: nat,J: nat,K: nat] :
% 0.19/0.46 ( ( ord_less_eq_nat @ I @ J )
% 0.19/0.46 => ( ( ord_less_eq_nat @ J @ K )
% 0.19/0.46 => ( ord_less_eq_nat @ I @ K ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % le_trans
% 0.19/0.46 thf(fact_14_eq__imp__le,axiom,
% 0.19/0.46 ! [M: nat,N: nat] :
% 0.19/0.46 ( ( M = N )
% 0.19/0.46 => ( ord_less_eq_nat @ M @ N ) ) ).
% 0.19/0.46
% 0.19/0.46 % eq_imp_le
% 0.19/0.46 thf(fact_15_le__antisym,axiom,
% 0.19/0.46 ! [M: nat,N: nat] :
% 0.19/0.46 ( ( ord_less_eq_nat @ M @ N )
% 0.19/0.46 => ( ( ord_less_eq_nat @ N @ M )
% 0.19/0.46 => ( M = N ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % le_antisym
% 0.19/0.46 thf(fact_16_nat__less__le,axiom,
% 0.19/0.46 ( ord_less_nat
% 0.19/0.46 = ( ^ [M2: nat,N2: nat] :
% 0.19/0.46 ( ( ord_less_eq_nat @ M2 @ N2 )
% 0.19/0.46 & ( M2 != N2 ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % nat_less_le
% 0.19/0.46 thf(fact_17_nat__neq__iff,axiom,
% 0.19/0.46 ! [M: nat,N: nat] :
% 0.19/0.46 ( ( M != N )
% 0.19/0.46 = ( ( ord_less_nat @ M @ N )
% 0.19/0.46 | ( ord_less_nat @ N @ M ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % nat_neq_iff
% 0.19/0.46 thf(fact_18_less__diff__iff,axiom,
% 0.19/0.46 ! [K: nat,M: nat,N: nat] :
% 0.19/0.46 ( ( ord_less_eq_nat @ K @ M )
% 0.19/0.46 => ( ( ord_less_eq_nat @ K @ N )
% 0.19/0.46 => ( ( ord_less_nat @ ( minus_minus_nat @ M @ K ) @ ( minus_minus_nat @ N @ K ) )
% 0.19/0.46 = ( ord_less_nat @ M @ N ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % less_diff_iff
% 0.19/0.46 thf(fact_19_less__not__refl,axiom,
% 0.19/0.46 ! [N: nat] :
% 0.19/0.46 ~ ( ord_less_nat @ N @ N ) ).
% 0.19/0.46
% 0.19/0.46 % less_not_refl
% 0.19/0.46 thf(fact_20_nat__le__linear,axiom,
% 0.19/0.46 ! [M: nat,N: nat] :
% 0.19/0.46 ( ( ord_less_eq_nat @ M @ N )
% 0.19/0.46 | ( ord_less_eq_nat @ N @ M ) ) ).
% 0.19/0.46
% 0.19/0.46 % nat_le_linear
% 0.19/0.46 thf(fact_21_diff__less__mono,axiom,
% 0.19/0.46 ! [A: nat,B: nat,C: nat] :
% 0.19/0.46 ( ( ord_less_nat @ A @ B )
% 0.19/0.46 => ( ( ord_less_eq_nat @ C @ A )
% 0.19/0.46 => ( ord_less_nat @ ( minus_minus_nat @ A @ C ) @ ( minus_minus_nat @ B @ C ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % diff_less_mono
% 0.19/0.46 thf(fact_22_less__not__refl2,axiom,
% 0.19/0.46 ! [N: nat,M: nat] :
% 0.19/0.46 ( ( ord_less_nat @ N @ M )
% 0.19/0.46 => ( M != N ) ) ).
% 0.19/0.46
% 0.19/0.46 % less_not_refl2
% 0.19/0.46 thf(fact_23_less__not__refl3,axiom,
% 0.19/0.46 ! [S: nat,T: nat] :
% 0.19/0.46 ( ( ord_less_nat @ S @ T )
% 0.19/0.46 => ( S != T ) ) ).
% 0.19/0.46
% 0.19/0.46 % less_not_refl3
% 0.19/0.46 thf(fact_24_less__imp__le__nat,axiom,
% 0.19/0.46 ! [M: nat,N: nat] :
% 0.19/0.46 ( ( ord_less_nat @ M @ N )
% 0.19/0.46 => ( ord_less_eq_nat @ M @ N ) ) ).
% 0.19/0.46
% 0.19/0.46 % less_imp_le_nat
% 0.19/0.46 thf(fact_25_less__irrefl__nat,axiom,
% 0.19/0.46 ! [N: nat] :
% 0.19/0.46 ~ ( ord_less_nat @ N @ N ) ).
% 0.19/0.46
% 0.19/0.46 % less_irrefl_nat
% 0.19/0.46 thf(fact_26_nat__less__induct,axiom,
% 0.19/0.46 ! [P: nat > $o,N: nat] :
% 0.19/0.46 ( ! [N3: nat] :
% 0.19/0.46 ( ! [M3: nat] :
% 0.19/0.46 ( ( ord_less_nat @ M3 @ N3 )
% 0.19/0.46 => ( P @ M3 ) )
% 0.19/0.46 => ( P @ N3 ) )
% 0.19/0.46 => ( P @ N ) ) ).
% 0.19/0.46
% 0.19/0.46 % nat_less_induct
% 0.19/0.46 thf(fact_27_infinite__descent,axiom,
% 0.19/0.46 ! [P: nat > $o,N: nat] :
% 0.19/0.46 ( ! [N3: nat] :
% 0.19/0.46 ( ~ ( P @ N3 )
% 0.19/0.46 => ? [M3: nat] :
% 0.19/0.46 ( ( ord_less_nat @ M3 @ N3 )
% 0.19/0.46 & ~ ( P @ M3 ) ) )
% 0.19/0.46 => ( P @ N ) ) ).
% 0.19/0.46
% 0.19/0.46 % infinite_descent
% 0.19/0.46 thf(fact_28_le__eq__less__or__eq,axiom,
% 0.19/0.46 ( ord_less_eq_nat
% 0.19/0.46 = ( ^ [M2: nat,N2: nat] :
% 0.19/0.46 ( ( ord_less_nat @ M2 @ N2 )
% 0.19/0.46 | ( M2 = N2 ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % le_eq_less_or_eq
% 0.19/0.46 thf(fact_29_less__or__eq__imp__le,axiom,
% 0.19/0.46 ! [M: nat,N: nat] :
% 0.19/0.46 ( ( ( ord_less_nat @ M @ N )
% 0.19/0.46 | ( M = N ) )
% 0.19/0.46 => ( ord_less_eq_nat @ M @ N ) ) ).
% 0.19/0.46
% 0.19/0.46 % less_or_eq_imp_le
% 0.19/0.46 thf(fact_30_linorder__neqE__nat,axiom,
% 0.19/0.46 ! [X: nat,Y: nat] :
% 0.19/0.46 ( ( X != Y )
% 0.19/0.46 => ( ~ ( ord_less_nat @ X @ Y )
% 0.19/0.46 => ( ord_less_nat @ Y @ X ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % linorder_neqE_nat
% 0.19/0.46 thf(fact_31_Nat_Oex__has__greatest__nat,axiom,
% 0.19/0.46 ! [P: nat > $o,K: nat,B: nat] :
% 0.19/0.46 ( ( P @ K )
% 0.19/0.46 => ( ! [Y2: nat] :
% 0.19/0.46 ( ( P @ Y2 )
% 0.19/0.46 => ( ord_less_eq_nat @ Y2 @ B ) )
% 0.19/0.46 => ? [X2: nat] :
% 0.19/0.46 ( ( P @ X2 )
% 0.19/0.46 & ! [Y3: nat] :
% 0.19/0.46 ( ( P @ Y3 )
% 0.19/0.46 => ( ord_less_eq_nat @ Y3 @ X2 ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % Nat.ex_has_greatest_nat
% 0.19/0.46 thf(fact_32_le__neq__implies__less,axiom,
% 0.19/0.46 ! [M: nat,N: nat] :
% 0.19/0.46 ( ( ord_less_eq_nat @ M @ N )
% 0.19/0.46 => ( ( M != N )
% 0.19/0.46 => ( ord_less_nat @ M @ N ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % le_neq_implies_less
% 0.19/0.46 thf(fact_33_less__mono__imp__le__mono,axiom,
% 0.19/0.46 ! [F: nat > nat,I: nat,J: nat] :
% 0.19/0.46 ( ! [I2: nat,J2: nat] :
% 0.19/0.46 ( ( ord_less_nat @ I2 @ J2 )
% 0.19/0.46 => ( ord_less_nat @ ( F @ I2 ) @ ( F @ J2 ) ) )
% 0.19/0.46 => ( ( ord_less_eq_nat @ I @ J )
% 0.19/0.46 => ( ord_less_eq_nat @ ( F @ I ) @ ( F @ J ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % less_mono_imp_le_mono
% 0.19/0.46 thf(fact_34_length__induct,axiom,
% 0.19/0.46 ! [P: list_c1059388851t_unit > $o,Xs: list_c1059388851t_unit] :
% 0.19/0.46 ( ! [Xs2: list_c1059388851t_unit] :
% 0.19/0.46 ( ! [Ys2: list_c1059388851t_unit] :
% 0.19/0.46 ( ( ord_less_nat @ ( size_s1406904903t_unit @ Ys2 ) @ ( size_s1406904903t_unit @ Xs2 ) )
% 0.19/0.46 => ( P @ Ys2 ) )
% 0.19/0.46 => ( P @ Xs2 ) )
% 0.19/0.46 => ( P @ Xs ) ) ).
% 0.19/0.46
% 0.19/0.46 % length_induct
% 0.19/0.46 thf(fact_35_less__imp__diff__less,axiom,
% 0.19/0.46 ! [J: nat,K: nat,N: nat] :
% 0.19/0.46 ( ( ord_less_nat @ J @ K )
% 0.19/0.46 => ( ord_less_nat @ ( minus_minus_nat @ J @ N ) @ K ) ) ).
% 0.19/0.46
% 0.19/0.46 % less_imp_diff_less
% 0.19/0.46 thf(fact_36_diff__less__mono2,axiom,
% 0.19/0.46 ! [M: nat,N: nat,L: nat] :
% 0.19/0.46 ( ( ord_less_nat @ M @ N )
% 0.19/0.46 => ( ( ord_less_nat @ M @ L )
% 0.19/0.46 => ( ord_less_nat @ ( minus_minus_nat @ L @ N ) @ ( minus_minus_nat @ L @ M ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % diff_less_mono2
% 0.19/0.46 thf(fact_37_diff__le__mono2,axiom,
% 0.19/0.46 ! [M: nat,N: nat,L: nat] :
% 0.19/0.46 ( ( ord_less_eq_nat @ M @ N )
% 0.19/0.46 => ( ord_less_eq_nat @ ( minus_minus_nat @ L @ N ) @ ( minus_minus_nat @ L @ M ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % diff_le_mono2
% 0.19/0.46 thf(fact_38_le__diff__iff_H,axiom,
% 0.19/0.46 ! [A: nat,C: nat,B: nat] :
% 0.19/0.46 ( ( ord_less_eq_nat @ A @ C )
% 0.19/0.46 => ( ( ord_less_eq_nat @ B @ C )
% 0.19/0.46 => ( ( ord_less_eq_nat @ ( minus_minus_nat @ C @ A ) @ ( minus_minus_nat @ C @ B ) )
% 0.19/0.46 = ( ord_less_eq_nat @ B @ A ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % le_diff_iff'
% 0.19/0.46 thf(fact_39_diff__le__self,axiom,
% 0.19/0.46 ! [M: nat,N: nat] : ( ord_less_eq_nat @ ( minus_minus_nat @ M @ N ) @ M ) ).
% 0.19/0.46
% 0.19/0.46 % diff_le_self
% 0.19/0.46 thf(fact_40_diff__le__mono,axiom,
% 0.19/0.46 ! [M: nat,N: nat,L: nat] :
% 0.19/0.46 ( ( ord_less_eq_nat @ M @ N )
% 0.19/0.46 => ( ord_less_eq_nat @ ( minus_minus_nat @ M @ L ) @ ( minus_minus_nat @ N @ L ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % diff_le_mono
% 0.19/0.46 thf(fact_41_Nat_Odiff__diff__eq,axiom,
% 0.19/0.46 ! [K: nat,M: nat,N: nat] :
% 0.19/0.46 ( ( ord_less_eq_nat @ K @ M )
% 0.19/0.46 => ( ( ord_less_eq_nat @ K @ N )
% 0.19/0.46 => ( ( minus_minus_nat @ ( minus_minus_nat @ M @ K ) @ ( minus_minus_nat @ N @ K ) )
% 0.19/0.46 = ( minus_minus_nat @ M @ N ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % Nat.diff_diff_eq
% 0.19/0.46 thf(fact_42_le__diff__iff,axiom,
% 0.19/0.46 ! [K: nat,M: nat,N: nat] :
% 0.19/0.46 ( ( ord_less_eq_nat @ K @ M )
% 0.19/0.46 => ( ( ord_less_eq_nat @ K @ N )
% 0.19/0.46 => ( ( ord_less_eq_nat @ ( minus_minus_nat @ M @ K ) @ ( minus_minus_nat @ N @ K ) )
% 0.19/0.46 = ( ord_less_eq_nat @ M @ N ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % le_diff_iff
% 0.19/0.46 thf(fact_43_eq__diff__iff,axiom,
% 0.19/0.46 ! [K: nat,M: nat,N: nat] :
% 0.19/0.46 ( ( ord_less_eq_nat @ K @ M )
% 0.19/0.46 => ( ( ord_less_eq_nat @ K @ N )
% 0.19/0.46 => ( ( ( minus_minus_nat @ M @ K )
% 0.19/0.46 = ( minus_minus_nat @ N @ K ) )
% 0.19/0.46 = ( M = N ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % eq_diff_iff
% 0.19/0.46 thf(fact_44_list__eq__iff__nth__eq,axiom,
% 0.19/0.46 ( ( ^ [Y4: list_c1059388851t_unit,Z: list_c1059388851t_unit] : ( Y4 = Z ) )
% 0.19/0.46 = ( ^ [Xs3: list_c1059388851t_unit,Ys3: list_c1059388851t_unit] :
% 0.19/0.46 ( ( ( size_s1406904903t_unit @ Xs3 )
% 0.19/0.46 = ( size_s1406904903t_unit @ Ys3 ) )
% 0.19/0.46 & ! [I3: nat] :
% 0.19/0.46 ( ( ord_less_nat @ I3 @ ( size_s1406904903t_unit @ Xs3 ) )
% 0.19/0.46 => ( ( nth_co1649820636t_unit @ Xs3 @ I3 )
% 0.19/0.46 = ( nth_co1649820636t_unit @ Ys3 @ I3 ) ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % list_eq_iff_nth_eq
% 0.19/0.46 thf(fact_45_Skolem__list__nth,axiom,
% 0.19/0.46 ! [K: nat,P: nat > config256849571t_unit > $o] :
% 0.19/0.46 ( ( ! [I3: nat] :
% 0.19/0.46 ( ( ord_less_nat @ I3 @ K )
% 0.19/0.46 => ? [X3: config256849571t_unit] : ( P @ I3 @ X3 ) ) )
% 0.19/0.46 = ( ? [Xs3: list_c1059388851t_unit] :
% 0.19/0.46 ( ( ( size_s1406904903t_unit @ Xs3 )
% 0.19/0.46 = K )
% 0.19/0.46 & ! [I3: nat] :
% 0.19/0.46 ( ( ord_less_nat @ I3 @ K )
% 0.19/0.46 => ( P @ I3 @ ( nth_co1649820636t_unit @ Xs3 @ I3 ) ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % Skolem_list_nth
% 0.19/0.46 thf(fact_46_nth__equalityI,axiom,
% 0.19/0.46 ! [Xs: list_c1059388851t_unit,Ys: list_c1059388851t_unit] :
% 0.19/0.46 ( ( ( size_s1406904903t_unit @ Xs )
% 0.19/0.46 = ( size_s1406904903t_unit @ Ys ) )
% 0.19/0.46 => ( ! [I2: nat] :
% 0.19/0.46 ( ( ord_less_nat @ I2 @ ( size_s1406904903t_unit @ Xs ) )
% 0.19/0.46 => ( ( nth_co1649820636t_unit @ Xs @ I2 )
% 0.19/0.46 = ( nth_co1649820636t_unit @ Ys @ I2 ) ) )
% 0.19/0.46 => ( Xs = Ys ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % nth_equalityI
% 0.19/0.46 thf(fact_47_one__reorient,axiom,
% 0.19/0.46 ! [X: nat] :
% 0.19/0.46 ( ( one_one_nat = X )
% 0.19/0.46 = ( X = one_one_nat ) ) ).
% 0.19/0.46
% 0.19/0.46 % one_reorient
% 0.19/0.46 thf(fact_48_diff__right__commute,axiom,
% 0.19/0.46 ! [A: nat,C: nat,B: nat] :
% 0.19/0.46 ( ( minus_minus_nat @ ( minus_minus_nat @ A @ C ) @ B )
% 0.19/0.46 = ( minus_minus_nat @ ( minus_minus_nat @ A @ B ) @ C ) ) ).
% 0.19/0.46
% 0.19/0.46 % diff_right_commute
% 0.19/0.46 thf(fact_49_Length,axiom,
% 0.19/0.46 ! [N4: nat] : ( ord_less_eq_nat @ ( plus_plus_nat @ N4 @ one_one_nat ) @ ( size_s1406904903t_unit @ ( fe @ N4 ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % Length
% 0.19/0.46 thf(fact_50_AllArePrefixesExec,axiom,
% 0.19/0.46 ! [M3: nat,N4: nat] :
% 0.19/0.46 ( ( ord_less_nat @ M3 @ N4 )
% 0.19/0.46 => ( prefix1615116500t_unit @ ( fe @ M3 ) @ ( fe @ N4 ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % AllArePrefixesExec
% 0.19/0.46 thf(fact_51_order__refl,axiom,
% 0.19/0.46 ! [X: nat] : ( ord_less_eq_nat @ X @ X ) ).
% 0.19/0.46
% 0.19/0.46 % order_refl
% 0.19/0.46 thf(fact_52_nat__descend__induct,axiom,
% 0.19/0.46 ! [N: nat,P: nat > $o,M: nat] :
% 0.19/0.46 ( ! [K2: nat] :
% 0.19/0.46 ( ( ord_less_nat @ N @ K2 )
% 0.19/0.46 => ( P @ K2 ) )
% 0.19/0.46 => ( ! [K2: nat] :
% 0.19/0.46 ( ( ord_less_eq_nat @ K2 @ N )
% 0.19/0.46 => ( ! [I4: nat] :
% 0.19/0.46 ( ( ord_less_nat @ K2 @ I4 )
% 0.19/0.46 => ( P @ I4 ) )
% 0.19/0.46 => ( P @ K2 ) ) )
% 0.19/0.46 => ( P @ M ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % nat_descend_induct
% 0.19/0.46 thf(fact_53_less__numeral__extra_I4_J,axiom,
% 0.19/0.46 ~ ( ord_less_nat @ one_one_nat @ one_one_nat ) ).
% 0.19/0.46
% 0.19/0.46 % less_numeral_extra(4)
% 0.19/0.46 thf(fact_54_le__numeral__extra_I4_J,axiom,
% 0.19/0.46 ord_less_eq_nat @ one_one_nat @ one_one_nat ).
% 0.19/0.46
% 0.19/0.46 % le_numeral_extra(4)
% 0.19/0.46 thf(fact_55_minf_I8_J,axiom,
% 0.19/0.46 ! [T: nat] :
% 0.19/0.46 ? [Z2: nat] :
% 0.19/0.46 ! [X4: nat] :
% 0.19/0.46 ( ( ord_less_nat @ X4 @ Z2 )
% 0.19/0.46 => ~ ( ord_less_eq_nat @ T @ X4 ) ) ).
% 0.19/0.46
% 0.19/0.46 % minf(8)
% 0.19/0.46 thf(fact_56_minf_I6_J,axiom,
% 0.19/0.46 ! [T: nat] :
% 0.19/0.46 ? [Z2: nat] :
% 0.19/0.46 ! [X4: nat] :
% 0.19/0.46 ( ( ord_less_nat @ X4 @ Z2 )
% 0.19/0.46 => ( ord_less_eq_nat @ X4 @ T ) ) ).
% 0.19/0.46
% 0.19/0.46 % minf(6)
% 0.19/0.46 thf(fact_57_pinf_I8_J,axiom,
% 0.19/0.46 ! [T: nat] :
% 0.19/0.46 ? [Z2: nat] :
% 0.19/0.46 ! [X4: nat] :
% 0.19/0.46 ( ( ord_less_nat @ Z2 @ X4 )
% 0.19/0.46 => ( ord_less_eq_nat @ T @ X4 ) ) ).
% 0.19/0.46
% 0.19/0.46 % pinf(8)
% 0.19/0.46 thf(fact_58_pinf_I6_J,axiom,
% 0.19/0.46 ! [T: nat] :
% 0.19/0.46 ? [Z2: nat] :
% 0.19/0.46 ! [X4: nat] :
% 0.19/0.46 ( ( ord_less_nat @ Z2 @ X4 )
% 0.19/0.46 => ~ ( ord_less_eq_nat @ X4 @ T ) ) ).
% 0.19/0.46
% 0.19/0.46 % pinf(6)
% 0.19/0.46 thf(fact_59_add__left__cancel,axiom,
% 0.19/0.46 ! [A: nat,B: nat,C: nat] :
% 0.19/0.46 ( ( ( plus_plus_nat @ A @ B )
% 0.19/0.46 = ( plus_plus_nat @ A @ C ) )
% 0.19/0.46 = ( B = C ) ) ).
% 0.19/0.46
% 0.19/0.46 % add_left_cancel
% 0.19/0.46 thf(fact_60_add__right__cancel,axiom,
% 0.19/0.46 ! [B: nat,A: nat,C: nat] :
% 0.19/0.46 ( ( ( plus_plus_nat @ B @ A )
% 0.19/0.46 = ( plus_plus_nat @ C @ A ) )
% 0.19/0.46 = ( B = C ) ) ).
% 0.19/0.46
% 0.19/0.46 % add_right_cancel
% 0.19/0.46 thf(fact_61_add__le__cancel__right,axiom,
% 0.19/0.46 ! [A: nat,C: nat,B: nat] :
% 0.19/0.46 ( ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) )
% 0.19/0.46 = ( ord_less_eq_nat @ A @ B ) ) ).
% 0.19/0.46
% 0.19/0.46 % add_le_cancel_right
% 0.19/0.46 thf(fact_62_add__le__cancel__left,axiom,
% 0.19/0.46 ! [C: nat,A: nat,B: nat] :
% 0.19/0.46 ( ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) )
% 0.19/0.46 = ( ord_less_eq_nat @ A @ B ) ) ).
% 0.19/0.46
% 0.19/0.46 % add_le_cancel_left
% 0.19/0.46 thf(fact_63_add__less__cancel__right,axiom,
% 0.19/0.46 ! [A: nat,C: nat,B: nat] :
% 0.19/0.46 ( ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) )
% 0.19/0.46 = ( ord_less_nat @ A @ B ) ) ).
% 0.19/0.46
% 0.19/0.46 % add_less_cancel_right
% 0.19/0.46 thf(fact_64_add__less__cancel__left,axiom,
% 0.19/0.46 ! [C: nat,A: nat,B: nat] :
% 0.19/0.46 ( ( ord_less_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) )
% 0.19/0.46 = ( ord_less_nat @ A @ B ) ) ).
% 0.19/0.46
% 0.19/0.46 % add_less_cancel_left
% 0.19/0.46 thf(fact_65_add__diff__cancel__left,axiom,
% 0.19/0.46 ! [C: nat,A: nat,B: nat] :
% 0.19/0.46 ( ( minus_minus_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) )
% 0.19/0.46 = ( minus_minus_nat @ A @ B ) ) ).
% 0.19/0.46
% 0.19/0.46 % add_diff_cancel_left
% 0.19/0.46 thf(fact_66_add__diff__cancel__left_H,axiom,
% 0.19/0.46 ! [A: nat,B: nat] :
% 0.19/0.46 ( ( minus_minus_nat @ ( plus_plus_nat @ A @ B ) @ A )
% 0.19/0.46 = B ) ).
% 0.19/0.46
% 0.19/0.46 % add_diff_cancel_left'
% 0.19/0.46 thf(fact_67_add__diff__cancel__right,axiom,
% 0.19/0.46 ! [A: nat,C: nat,B: nat] :
% 0.19/0.46 ( ( minus_minus_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) )
% 0.19/0.46 = ( minus_minus_nat @ A @ B ) ) ).
% 0.19/0.46
% 0.19/0.46 % add_diff_cancel_right
% 0.19/0.46 thf(fact_68_add__diff__cancel__right_H,axiom,
% 0.19/0.46 ! [A: nat,B: nat] :
% 0.19/0.46 ( ( minus_minus_nat @ ( plus_plus_nat @ A @ B ) @ B )
% 0.19/0.46 = A ) ).
% 0.19/0.46
% 0.19/0.46 % add_diff_cancel_right'
% 0.19/0.46 thf(fact_69_nat__add__left__cancel__less,axiom,
% 0.19/0.46 ! [K: nat,M: nat,N: nat] :
% 0.19/0.46 ( ( ord_less_nat @ ( plus_plus_nat @ K @ M ) @ ( plus_plus_nat @ K @ N ) )
% 0.19/0.46 = ( ord_less_nat @ M @ N ) ) ).
% 0.19/0.46
% 0.19/0.46 % nat_add_left_cancel_less
% 0.19/0.46 thf(fact_70_nat__add__left__cancel__le,axiom,
% 0.19/0.46 ! [K: nat,M: nat,N: nat] :
% 0.19/0.46 ( ( ord_less_eq_nat @ ( plus_plus_nat @ K @ M ) @ ( plus_plus_nat @ K @ N ) )
% 0.19/0.46 = ( ord_less_eq_nat @ M @ N ) ) ).
% 0.19/0.46
% 0.19/0.46 % nat_add_left_cancel_le
% 0.19/0.46 thf(fact_71_diff__diff__left,axiom,
% 0.19/0.46 ! [I: nat,J: nat,K: nat] :
% 0.19/0.46 ( ( minus_minus_nat @ ( minus_minus_nat @ I @ J ) @ K )
% 0.19/0.46 = ( minus_minus_nat @ I @ ( plus_plus_nat @ J @ K ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % diff_diff_left
% 0.19/0.46 thf(fact_72_Nat_Odiff__diff__right,axiom,
% 0.19/0.46 ! [K: nat,J: nat,I: nat] :
% 0.19/0.46 ( ( ord_less_eq_nat @ K @ J )
% 0.19/0.46 => ( ( minus_minus_nat @ I @ ( minus_minus_nat @ J @ K ) )
% 0.19/0.46 = ( minus_minus_nat @ ( plus_plus_nat @ I @ K ) @ J ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % Nat.diff_diff_right
% 0.19/0.46 thf(fact_73_Nat_Oadd__diff__assoc2,axiom,
% 0.19/0.46 ! [K: nat,J: nat,I: nat] :
% 0.19/0.46 ( ( ord_less_eq_nat @ K @ J )
% 0.19/0.46 => ( ( plus_plus_nat @ ( minus_minus_nat @ J @ K ) @ I )
% 0.19/0.46 = ( minus_minus_nat @ ( plus_plus_nat @ J @ I ) @ K ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % Nat.add_diff_assoc2
% 0.19/0.46 thf(fact_74_Nat_Oadd__diff__assoc,axiom,
% 0.19/0.46 ! [K: nat,J: nat,I: nat] :
% 0.19/0.46 ( ( ord_less_eq_nat @ K @ J )
% 0.19/0.46 => ( ( plus_plus_nat @ I @ ( minus_minus_nat @ J @ K ) )
% 0.19/0.46 = ( minus_minus_nat @ ( plus_plus_nat @ I @ J ) @ K ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % Nat.add_diff_assoc
% 0.19/0.46 thf(fact_75_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
% 0.19/0.46 ! [A: nat,B: nat,C: nat] :
% 0.19/0.46 ( ( plus_plus_nat @ ( plus_plus_nat @ A @ B ) @ C )
% 0.19/0.46 = ( plus_plus_nat @ A @ ( plus_plus_nat @ B @ C ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % ab_semigroup_add_class.add_ac(1)
% 0.19/0.46 thf(fact_76_add__mono__thms__linordered__semiring_I4_J,axiom,
% 0.19/0.46 ! [I: nat,J: nat,K: nat,L: nat] :
% 0.19/0.46 ( ( ( I = J )
% 0.19/0.46 & ( K = L ) )
% 0.19/0.46 => ( ( plus_plus_nat @ I @ K )
% 0.19/0.46 = ( plus_plus_nat @ J @ L ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % add_mono_thms_linordered_semiring(4)
% 0.19/0.46 thf(fact_77_group__cancel_Oadd1,axiom,
% 0.19/0.46 ! [A2: nat,K: nat,A: nat,B: nat] :
% 0.19/0.46 ( ( A2
% 0.19/0.46 = ( plus_plus_nat @ K @ A ) )
% 0.19/0.46 => ( ( plus_plus_nat @ A2 @ B )
% 0.19/0.46 = ( plus_plus_nat @ K @ ( plus_plus_nat @ A @ B ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % group_cancel.add1
% 0.19/0.46 thf(fact_78_group__cancel_Oadd2,axiom,
% 0.19/0.46 ! [B2: nat,K: nat,B: nat,A: nat] :
% 0.19/0.46 ( ( B2
% 0.19/0.46 = ( plus_plus_nat @ K @ B ) )
% 0.19/0.46 => ( ( plus_plus_nat @ A @ B2 )
% 0.19/0.46 = ( plus_plus_nat @ K @ ( plus_plus_nat @ A @ B ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % group_cancel.add2
% 0.19/0.46 thf(fact_79_add_Oassoc,axiom,
% 0.19/0.46 ! [A: nat,B: nat,C: nat] :
% 0.19/0.46 ( ( plus_plus_nat @ ( plus_plus_nat @ A @ B ) @ C )
% 0.19/0.46 = ( plus_plus_nat @ A @ ( plus_plus_nat @ B @ C ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % add.assoc
% 0.19/0.46 thf(fact_80_add_Ocommute,axiom,
% 0.19/0.46 ( plus_plus_nat
% 0.19/0.46 = ( ^ [A3: nat,B3: nat] : ( plus_plus_nat @ B3 @ A3 ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % add.commute
% 0.19/0.46 thf(fact_81_add_Oleft__commute,axiom,
% 0.19/0.46 ! [B: nat,A: nat,C: nat] :
% 0.19/0.46 ( ( plus_plus_nat @ B @ ( plus_plus_nat @ A @ C ) )
% 0.19/0.46 = ( plus_plus_nat @ A @ ( plus_plus_nat @ B @ C ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % add.left_commute
% 0.19/0.46 thf(fact_82_add__left__imp__eq,axiom,
% 0.19/0.46 ! [A: nat,B: nat,C: nat] :
% 0.19/0.46 ( ( ( plus_plus_nat @ A @ B )
% 0.19/0.46 = ( plus_plus_nat @ A @ C ) )
% 0.19/0.46 => ( B = C ) ) ).
% 0.19/0.46
% 0.19/0.46 % add_left_imp_eq
% 0.19/0.46 thf(fact_83_add__right__imp__eq,axiom,
% 0.19/0.46 ! [B: nat,A: nat,C: nat] :
% 0.19/0.46 ( ( ( plus_plus_nat @ B @ A )
% 0.19/0.46 = ( plus_plus_nat @ C @ A ) )
% 0.19/0.46 => ( B = C ) ) ).
% 0.19/0.46
% 0.19/0.46 % add_right_imp_eq
% 0.19/0.46 thf(fact_84_add__le__imp__le__right,axiom,
% 0.19/0.46 ! [A: nat,C: nat,B: nat] :
% 0.19/0.46 ( ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) )
% 0.19/0.46 => ( ord_less_eq_nat @ A @ B ) ) ).
% 0.19/0.46
% 0.19/0.46 % add_le_imp_le_right
% 0.19/0.46 thf(fact_85_add__le__imp__le__left,axiom,
% 0.19/0.46 ! [C: nat,A: nat,B: nat] :
% 0.19/0.46 ( ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) )
% 0.19/0.46 => ( ord_less_eq_nat @ A @ B ) ) ).
% 0.19/0.46
% 0.19/0.46 % add_le_imp_le_left
% 0.19/0.46 thf(fact_86_le__iff__add,axiom,
% 0.19/0.46 ( ord_less_eq_nat
% 0.19/0.46 = ( ^ [A3: nat,B3: nat] :
% 0.19/0.46 ? [C2: nat] :
% 0.19/0.46 ( B3
% 0.19/0.46 = ( plus_plus_nat @ A3 @ C2 ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % le_iff_add
% 0.19/0.46 thf(fact_87_add__right__mono,axiom,
% 0.19/0.46 ! [A: nat,B: nat,C: nat] :
% 0.19/0.46 ( ( ord_less_eq_nat @ A @ B )
% 0.19/0.46 => ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % add_right_mono
% 0.19/0.46 thf(fact_88_less__eqE,axiom,
% 0.19/0.46 ! [A: nat,B: nat] :
% 0.19/0.46 ( ( ord_less_eq_nat @ A @ B )
% 0.19/0.46 => ~ ! [C3: nat] :
% 0.19/0.46 ( B
% 0.19/0.46 != ( plus_plus_nat @ A @ C3 ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % less_eqE
% 0.19/0.46 thf(fact_89_add__left__mono,axiom,
% 0.19/0.46 ! [A: nat,B: nat,C: nat] :
% 0.19/0.46 ( ( ord_less_eq_nat @ A @ B )
% 0.19/0.46 => ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % add_left_mono
% 0.19/0.46 thf(fact_90_add__mono,axiom,
% 0.19/0.46 ! [A: nat,B: nat,C: nat,D: nat] :
% 0.19/0.46 ( ( ord_less_eq_nat @ A @ B )
% 0.19/0.46 => ( ( ord_less_eq_nat @ C @ D )
% 0.19/0.46 => ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ D ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % add_mono
% 0.19/0.46 thf(fact_91_add__mono__thms__linordered__semiring_I1_J,axiom,
% 0.19/0.46 ! [I: nat,J: nat,K: nat,L: nat] :
% 0.19/0.46 ( ( ( ord_less_eq_nat @ I @ J )
% 0.19/0.46 & ( ord_less_eq_nat @ K @ L ) )
% 0.19/0.46 => ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % add_mono_thms_linordered_semiring(1)
% 0.19/0.46 thf(fact_92_add__mono__thms__linordered__semiring_I2_J,axiom,
% 0.19/0.46 ! [I: nat,J: nat,K: nat,L: nat] :
% 0.19/0.46 ( ( ( I = J )
% 0.19/0.46 & ( ord_less_eq_nat @ K @ L ) )
% 0.19/0.46 => ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % add_mono_thms_linordered_semiring(2)
% 0.19/0.46 thf(fact_93_add__mono__thms__linordered__semiring_I3_J,axiom,
% 0.19/0.46 ! [I: nat,J: nat,K: nat,L: nat] :
% 0.19/0.46 ( ( ( ord_less_eq_nat @ I @ J )
% 0.19/0.46 & ( K = L ) )
% 0.19/0.46 => ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % add_mono_thms_linordered_semiring(3)
% 0.19/0.46 thf(fact_94_add__less__imp__less__right,axiom,
% 0.19/0.46 ! [A: nat,C: nat,B: nat] :
% 0.19/0.46 ( ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) )
% 0.19/0.46 => ( ord_less_nat @ A @ B ) ) ).
% 0.19/0.46
% 0.19/0.46 % add_less_imp_less_right
% 0.19/0.46 thf(fact_95_add__less__imp__less__left,axiom,
% 0.19/0.46 ! [C: nat,A: nat,B: nat] :
% 0.19/0.46 ( ( ord_less_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) )
% 0.19/0.46 => ( ord_less_nat @ A @ B ) ) ).
% 0.19/0.46
% 0.19/0.46 % add_less_imp_less_left
% 0.19/0.46 thf(fact_96_add__strict__right__mono,axiom,
% 0.19/0.46 ! [A: nat,B: nat,C: nat] :
% 0.19/0.46 ( ( ord_less_nat @ A @ B )
% 0.19/0.46 => ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % add_strict_right_mono
% 0.19/0.46 thf(fact_97_add__strict__left__mono,axiom,
% 0.19/0.46 ! [A: nat,B: nat,C: nat] :
% 0.19/0.46 ( ( ord_less_nat @ A @ B )
% 0.19/0.46 => ( ord_less_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % add_strict_left_mono
% 0.19/0.46 thf(fact_98_add__strict__mono,axiom,
% 0.19/0.46 ! [A: nat,B: nat,C: nat,D: nat] :
% 0.19/0.46 ( ( ord_less_nat @ A @ B )
% 0.19/0.46 => ( ( ord_less_nat @ C @ D )
% 0.19/0.46 => ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ D ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % add_strict_mono
% 0.19/0.46 thf(fact_99_add__mono__thms__linordered__field_I1_J,axiom,
% 0.19/0.46 ! [I: nat,J: nat,K: nat,L: nat] :
% 0.19/0.46 ( ( ( ord_less_nat @ I @ J )
% 0.19/0.46 & ( K = L ) )
% 0.19/0.46 => ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % add_mono_thms_linordered_field(1)
% 0.19/0.46 thf(fact_100_add__mono__thms__linordered__field_I2_J,axiom,
% 0.19/0.46 ! [I: nat,J: nat,K: nat,L: nat] :
% 0.19/0.46 ( ( ( I = J )
% 0.19/0.46 & ( ord_less_nat @ K @ L ) )
% 0.19/0.46 => ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % add_mono_thms_linordered_field(2)
% 0.19/0.46 thf(fact_101_add__mono__thms__linordered__field_I5_J,axiom,
% 0.19/0.46 ! [I: nat,J: nat,K: nat,L: nat] :
% 0.19/0.46 ( ( ( ord_less_nat @ I @ J )
% 0.19/0.46 & ( ord_less_nat @ K @ L ) )
% 0.19/0.46 => ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % add_mono_thms_linordered_field(5)
% 0.19/0.46 thf(fact_102_diff__diff__add,axiom,
% 0.19/0.46 ! [A: nat,B: nat,C: nat] :
% 0.19/0.46 ( ( minus_minus_nat @ ( minus_minus_nat @ A @ B ) @ C )
% 0.19/0.46 = ( minus_minus_nat @ A @ ( plus_plus_nat @ B @ C ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % diff_diff_add
% 0.19/0.46 thf(fact_103_add__implies__diff,axiom,
% 0.19/0.46 ! [C: nat,B: nat,A: nat] :
% 0.19/0.46 ( ( ( plus_plus_nat @ C @ B )
% 0.19/0.46 = A )
% 0.19/0.46 => ( C
% 0.19/0.46 = ( minus_minus_nat @ A @ B ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % add_implies_diff
% 0.19/0.46 thf(fact_104_less__add__eq__less,axiom,
% 0.19/0.46 ! [K: nat,L: nat,M: nat,N: nat] :
% 0.19/0.46 ( ( ord_less_nat @ K @ L )
% 0.19/0.46 => ( ( ( plus_plus_nat @ M @ L )
% 0.19/0.46 = ( plus_plus_nat @ K @ N ) )
% 0.19/0.46 => ( ord_less_nat @ M @ N ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % less_add_eq_less
% 0.19/0.46 thf(fact_105_trans__less__add2,axiom,
% 0.19/0.46 ! [I: nat,J: nat,M: nat] :
% 0.19/0.46 ( ( ord_less_nat @ I @ J )
% 0.19/0.46 => ( ord_less_nat @ I @ ( plus_plus_nat @ M @ J ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % trans_less_add2
% 0.19/0.46 thf(fact_106_trans__less__add1,axiom,
% 0.19/0.46 ! [I: nat,J: nat,M: nat] :
% 0.19/0.46 ( ( ord_less_nat @ I @ J )
% 0.19/0.46 => ( ord_less_nat @ I @ ( plus_plus_nat @ J @ M ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % trans_less_add1
% 0.19/0.46 thf(fact_107_add__less__mono1,axiom,
% 0.19/0.46 ! [I: nat,J: nat,K: nat] :
% 0.19/0.46 ( ( ord_less_nat @ I @ J )
% 0.19/0.46 => ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ K ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % add_less_mono1
% 0.19/0.46 thf(fact_108_not__add__less2,axiom,
% 0.19/0.46 ! [J: nat,I: nat] :
% 0.19/0.46 ~ ( ord_less_nat @ ( plus_plus_nat @ J @ I ) @ I ) ).
% 0.19/0.46
% 0.19/0.46 % not_add_less2
% 0.19/0.46 thf(fact_109_not__add__less1,axiom,
% 0.19/0.46 ! [I: nat,J: nat] :
% 0.19/0.46 ~ ( ord_less_nat @ ( plus_plus_nat @ I @ J ) @ I ) ).
% 0.19/0.46
% 0.19/0.46 % not_add_less1
% 0.19/0.46 thf(fact_110_add__less__mono,axiom,
% 0.19/0.46 ! [I: nat,J: nat,K: nat,L: nat] :
% 0.19/0.46 ( ( ord_less_nat @ I @ J )
% 0.19/0.46 => ( ( ord_less_nat @ K @ L )
% 0.19/0.46 => ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % add_less_mono
% 0.19/0.46 thf(fact_111_add__lessD1,axiom,
% 0.19/0.46 ! [I: nat,J: nat,K: nat] :
% 0.19/0.46 ( ( ord_less_nat @ ( plus_plus_nat @ I @ J ) @ K )
% 0.19/0.46 => ( ord_less_nat @ I @ K ) ) ).
% 0.19/0.46
% 0.19/0.46 % add_lessD1
% 0.19/0.46 thf(fact_112_nat__le__iff__add,axiom,
% 0.19/0.46 ( ord_less_eq_nat
% 0.19/0.46 = ( ^ [M2: nat,N2: nat] :
% 0.19/0.46 ? [K3: nat] :
% 0.19/0.46 ( N2
% 0.19/0.46 = ( plus_plus_nat @ M2 @ K3 ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % nat_le_iff_add
% 0.19/0.46 thf(fact_113_trans__le__add2,axiom,
% 0.19/0.46 ! [I: nat,J: nat,M: nat] :
% 0.19/0.46 ( ( ord_less_eq_nat @ I @ J )
% 0.19/0.46 => ( ord_less_eq_nat @ I @ ( plus_plus_nat @ M @ J ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % trans_le_add2
% 0.19/0.46 thf(fact_114_trans__le__add1,axiom,
% 0.19/0.46 ! [I: nat,J: nat,M: nat] :
% 0.19/0.46 ( ( ord_less_eq_nat @ I @ J )
% 0.19/0.46 => ( ord_less_eq_nat @ I @ ( plus_plus_nat @ J @ M ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % trans_le_add1
% 0.19/0.46 thf(fact_115_add__le__mono1,axiom,
% 0.19/0.46 ! [I: nat,J: nat,K: nat] :
% 0.19/0.46 ( ( ord_less_eq_nat @ I @ J )
% 0.19/0.46 => ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ K ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % add_le_mono1
% 0.19/0.46 thf(fact_116_add__le__mono,axiom,
% 0.19/0.46 ! [I: nat,J: nat,K: nat,L: nat] :
% 0.19/0.46 ( ( ord_less_eq_nat @ I @ J )
% 0.19/0.46 => ( ( ord_less_eq_nat @ K @ L )
% 0.19/0.46 => ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % add_le_mono
% 0.19/0.46 thf(fact_117_le__Suc__ex,axiom,
% 0.19/0.46 ! [K: nat,L: nat] :
% 0.19/0.46 ( ( ord_less_eq_nat @ K @ L )
% 0.19/0.46 => ? [N3: nat] :
% 0.19/0.46 ( L
% 0.19/0.46 = ( plus_plus_nat @ K @ N3 ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % le_Suc_ex
% 0.19/0.46 thf(fact_118_add__leD2,axiom,
% 0.19/0.46 ! [M: nat,K: nat,N: nat] :
% 0.19/0.46 ( ( ord_less_eq_nat @ ( plus_plus_nat @ M @ K ) @ N )
% 0.19/0.46 => ( ord_less_eq_nat @ K @ N ) ) ).
% 0.19/0.46
% 0.19/0.46 % add_leD2
% 0.19/0.46 thf(fact_119_add__leD1,axiom,
% 0.19/0.46 ! [M: nat,K: nat,N: nat] :
% 0.19/0.46 ( ( ord_less_eq_nat @ ( plus_plus_nat @ M @ K ) @ N )
% 0.19/0.46 => ( ord_less_eq_nat @ M @ N ) ) ).
% 0.19/0.46
% 0.19/0.46 % add_leD1
% 0.19/0.46 thf(fact_120_le__add2,axiom,
% 0.19/0.46 ! [N: nat,M: nat] : ( ord_less_eq_nat @ N @ ( plus_plus_nat @ M @ N ) ) ).
% 0.19/0.46
% 0.19/0.46 % le_add2
% 0.19/0.46 thf(fact_121_le__add1,axiom,
% 0.19/0.46 ! [N: nat,M: nat] : ( ord_less_eq_nat @ N @ ( plus_plus_nat @ N @ M ) ) ).
% 0.19/0.46
% 0.19/0.46 % le_add1
% 0.19/0.46 thf(fact_122_add__leE,axiom,
% 0.19/0.46 ! [M: nat,K: nat,N: nat] :
% 0.19/0.46 ( ( ord_less_eq_nat @ ( plus_plus_nat @ M @ K ) @ N )
% 0.19/0.46 => ~ ( ( ord_less_eq_nat @ M @ N )
% 0.19/0.46 => ~ ( ord_less_eq_nat @ K @ N ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % add_leE
% 0.19/0.46 thf(fact_123_Nat_Odiff__cancel,axiom,
% 0.19/0.46 ! [K: nat,M: nat,N: nat] :
% 0.19/0.46 ( ( minus_minus_nat @ ( plus_plus_nat @ K @ M ) @ ( plus_plus_nat @ K @ N ) )
% 0.19/0.46 = ( minus_minus_nat @ M @ N ) ) ).
% 0.19/0.46
% 0.19/0.46 % Nat.diff_cancel
% 0.19/0.46 thf(fact_124_diff__cancel2,axiom,
% 0.19/0.46 ! [M: nat,K: nat,N: nat] :
% 0.19/0.46 ( ( minus_minus_nat @ ( plus_plus_nat @ M @ K ) @ ( plus_plus_nat @ N @ K ) )
% 0.19/0.46 = ( minus_minus_nat @ M @ N ) ) ).
% 0.19/0.46
% 0.19/0.46 % diff_cancel2
% 0.19/0.46 thf(fact_125_diff__add__inverse,axiom,
% 0.19/0.46 ! [N: nat,M: nat] :
% 0.19/0.46 ( ( minus_minus_nat @ ( plus_plus_nat @ N @ M ) @ N )
% 0.19/0.46 = M ) ).
% 0.19/0.46
% 0.19/0.46 % diff_add_inverse
% 0.19/0.46 thf(fact_126_diff__add__inverse2,axiom,
% 0.19/0.46 ! [M: nat,N: nat] :
% 0.19/0.46 ( ( minus_minus_nat @ ( plus_plus_nat @ M @ N ) @ N )
% 0.19/0.46 = M ) ).
% 0.19/0.46
% 0.19/0.46 % diff_add_inverse2
% 0.19/0.46 thf(fact_127_add__less__le__mono,axiom,
% 0.19/0.46 ! [A: nat,B: nat,C: nat,D: nat] :
% 0.19/0.46 ( ( ord_less_nat @ A @ B )
% 0.19/0.46 => ( ( ord_less_eq_nat @ C @ D )
% 0.19/0.46 => ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ D ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % add_less_le_mono
% 0.19/0.46 thf(fact_128_add__le__less__mono,axiom,
% 0.19/0.46 ! [A: nat,B: nat,C: nat,D: nat] :
% 0.19/0.46 ( ( ord_less_eq_nat @ A @ B )
% 0.19/0.46 => ( ( ord_less_nat @ C @ D )
% 0.19/0.46 => ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ D ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % add_le_less_mono
% 0.19/0.46 thf(fact_129_add__mono__thms__linordered__field_I3_J,axiom,
% 0.19/0.46 ! [I: nat,J: nat,K: nat,L: nat] :
% 0.19/0.46 ( ( ( ord_less_nat @ I @ J )
% 0.19/0.46 & ( ord_less_eq_nat @ K @ L ) )
% 0.19/0.46 => ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % add_mono_thms_linordered_field(3)
% 0.19/0.46 thf(fact_130_add__mono__thms__linordered__field_I4_J,axiom,
% 0.19/0.46 ! [I: nat,J: nat,K: nat,L: nat] :
% 0.19/0.46 ( ( ( ord_less_eq_nat @ I @ J )
% 0.19/0.46 & ( ord_less_nat @ K @ L ) )
% 0.19/0.46 => ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % add_mono_thms_linordered_field(4)
% 0.19/0.46 thf(fact_131_ordered__cancel__comm__monoid__diff__class_Ole__imp__diff__is__add,axiom,
% 0.19/0.46 ! [A: nat,B: nat,C: nat] :
% 0.19/0.46 ( ( ord_less_eq_nat @ A @ B )
% 0.19/0.46 => ( ( ord_less_eq_nat @ A @ B )
% 0.19/0.46 => ( ( ( minus_minus_nat @ B @ A )
% 0.19/0.46 = C )
% 0.19/0.46 = ( B
% 0.19/0.46 = ( plus_plus_nat @ C @ A ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % ordered_cancel_comm_monoid_diff_class.le_imp_diff_is_add
% 0.19/0.46 thf(fact_132_ordered__cancel__comm__monoid__diff__class_Oadd__diff__inverse,axiom,
% 0.19/0.46 ! [A: nat,B: nat] :
% 0.19/0.46 ( ( ord_less_eq_nat @ A @ B )
% 0.19/0.46 => ( ( plus_plus_nat @ A @ ( minus_minus_nat @ B @ A ) )
% 0.19/0.46 = B ) ) ).
% 0.19/0.46
% 0.19/0.46 % ordered_cancel_comm_monoid_diff_class.add_diff_inverse
% 0.19/0.46 thf(fact_133_ordered__cancel__comm__monoid__diff__class_Odiff__diff__right,axiom,
% 0.19/0.46 ! [A: nat,B: nat,C: nat] :
% 0.19/0.46 ( ( ord_less_eq_nat @ A @ B )
% 0.19/0.46 => ( ( minus_minus_nat @ C @ ( minus_minus_nat @ B @ A ) )
% 0.19/0.46 = ( minus_minus_nat @ ( plus_plus_nat @ C @ A ) @ B ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % ordered_cancel_comm_monoid_diff_class.diff_diff_right
% 0.19/0.46 thf(fact_134_ordered__cancel__comm__monoid__diff__class_Odiff__add__assoc2,axiom,
% 0.19/0.46 ! [A: nat,B: nat,C: nat] :
% 0.19/0.46 ( ( ord_less_eq_nat @ A @ B )
% 0.19/0.46 => ( ( minus_minus_nat @ ( plus_plus_nat @ B @ C ) @ A )
% 0.19/0.46 = ( plus_plus_nat @ ( minus_minus_nat @ B @ A ) @ C ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % ordered_cancel_comm_monoid_diff_class.diff_add_assoc2
% 0.19/0.46 thf(fact_135_ordered__cancel__comm__monoid__diff__class_Oadd__diff__assoc2,axiom,
% 0.19/0.46 ! [A: nat,B: nat,C: nat] :
% 0.19/0.46 ( ( ord_less_eq_nat @ A @ B )
% 0.19/0.46 => ( ( plus_plus_nat @ ( minus_minus_nat @ B @ A ) @ C )
% 0.19/0.46 = ( minus_minus_nat @ ( plus_plus_nat @ B @ C ) @ A ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % ordered_cancel_comm_monoid_diff_class.add_diff_assoc2
% 0.19/0.46 thf(fact_136_ordered__cancel__comm__monoid__diff__class_Odiff__add__assoc,axiom,
% 0.19/0.46 ! [A: nat,B: nat,C: nat] :
% 0.19/0.46 ( ( ord_less_eq_nat @ A @ B )
% 0.19/0.46 => ( ( minus_minus_nat @ ( plus_plus_nat @ C @ B ) @ A )
% 0.19/0.46 = ( plus_plus_nat @ C @ ( minus_minus_nat @ B @ A ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % ordered_cancel_comm_monoid_diff_class.diff_add_assoc
% 0.19/0.46 thf(fact_137_ordered__cancel__comm__monoid__diff__class_Oadd__diff__assoc,axiom,
% 0.19/0.46 ! [A: nat,B: nat,C: nat] :
% 0.19/0.46 ( ( ord_less_eq_nat @ A @ B )
% 0.19/0.46 => ( ( plus_plus_nat @ C @ ( minus_minus_nat @ B @ A ) )
% 0.19/0.46 = ( minus_minus_nat @ ( plus_plus_nat @ C @ B ) @ A ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % ordered_cancel_comm_monoid_diff_class.add_diff_assoc
% 0.19/0.46 thf(fact_138_ordered__cancel__comm__monoid__diff__class_Ole__diff__conv2,axiom,
% 0.19/0.46 ! [A: nat,B: nat,C: nat] :
% 0.19/0.46 ( ( ord_less_eq_nat @ A @ B )
% 0.19/0.46 => ( ( ord_less_eq_nat @ C @ ( minus_minus_nat @ B @ A ) )
% 0.19/0.46 = ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A ) @ B ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % ordered_cancel_comm_monoid_diff_class.le_diff_conv2
% 0.19/0.46 thf(fact_139_le__add__diff,axiom,
% 0.19/0.46 ! [A: nat,B: nat,C: nat] :
% 0.19/0.46 ( ( ord_less_eq_nat @ A @ B )
% 0.19/0.46 => ( ord_less_eq_nat @ C @ ( minus_minus_nat @ ( plus_plus_nat @ B @ C ) @ A ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % le_add_diff
% 0.19/0.46 thf(fact_140_diff__add,axiom,
% 0.19/0.46 ! [A: nat,B: nat] :
% 0.19/0.46 ( ( ord_less_eq_nat @ A @ B )
% 0.19/0.46 => ( ( plus_plus_nat @ ( minus_minus_nat @ B @ A ) @ A )
% 0.19/0.46 = B ) ) ).
% 0.19/0.46
% 0.19/0.46 % diff_add
% 0.19/0.46 thf(fact_141_mono__nat__linear__lb,axiom,
% 0.19/0.46 ! [F: nat > nat,M: nat,K: nat] :
% 0.19/0.46 ( ! [M4: nat,N3: nat] :
% 0.19/0.46 ( ( ord_less_nat @ M4 @ N3 )
% 0.19/0.46 => ( ord_less_nat @ ( F @ M4 ) @ ( F @ N3 ) ) )
% 0.19/0.46 => ( ord_less_eq_nat @ ( plus_plus_nat @ ( F @ M ) @ K ) @ ( F @ ( plus_plus_nat @ M @ K ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % mono_nat_linear_lb
% 0.19/0.46 thf(fact_142_add__diff__inverse__nat,axiom,
% 0.19/0.46 ! [M: nat,N: nat] :
% 0.19/0.46 ( ~ ( ord_less_nat @ M @ N )
% 0.19/0.46 => ( ( plus_plus_nat @ N @ ( minus_minus_nat @ M @ N ) )
% 0.19/0.46 = M ) ) ).
% 0.19/0.46
% 0.19/0.46 % add_diff_inverse_nat
% 0.19/0.46 thf(fact_143_less__diff__conv,axiom,
% 0.19/0.46 ! [I: nat,J: nat,K: nat] :
% 0.19/0.46 ( ( ord_less_nat @ I @ ( minus_minus_nat @ J @ K ) )
% 0.19/0.46 = ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ J ) ) ).
% 0.19/0.46
% 0.19/0.46 % less_diff_conv
% 0.19/0.46 thf(fact_144_Nat_Ole__imp__diff__is__add,axiom,
% 0.19/0.46 ! [I: nat,J: nat,K: nat] :
% 0.19/0.46 ( ( ord_less_eq_nat @ I @ J )
% 0.19/0.46 => ( ( ( minus_minus_nat @ J @ I )
% 0.19/0.46 = K )
% 0.19/0.46 = ( J
% 0.19/0.46 = ( plus_plus_nat @ K @ I ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % Nat.le_imp_diff_is_add
% 0.19/0.46 thf(fact_145_Nat_Odiff__add__assoc2,axiom,
% 0.19/0.46 ! [K: nat,J: nat,I: nat] :
% 0.19/0.46 ( ( ord_less_eq_nat @ K @ J )
% 0.19/0.46 => ( ( minus_minus_nat @ ( plus_plus_nat @ J @ I ) @ K )
% 0.19/0.46 = ( plus_plus_nat @ ( minus_minus_nat @ J @ K ) @ I ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % Nat.diff_add_assoc2
% 0.19/0.46 thf(fact_146_Nat_Odiff__add__assoc,axiom,
% 0.19/0.46 ! [K: nat,J: nat,I: nat] :
% 0.19/0.46 ( ( ord_less_eq_nat @ K @ J )
% 0.19/0.46 => ( ( minus_minus_nat @ ( plus_plus_nat @ I @ J ) @ K )
% 0.19/0.46 = ( plus_plus_nat @ I @ ( minus_minus_nat @ J @ K ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % Nat.diff_add_assoc
% 0.19/0.46 thf(fact_147_Nat_Ole__diff__conv2,axiom,
% 0.19/0.46 ! [K: nat,J: nat,I: nat] :
% 0.19/0.46 ( ( ord_less_eq_nat @ K @ J )
% 0.19/0.46 => ( ( ord_less_eq_nat @ I @ ( minus_minus_nat @ J @ K ) )
% 0.19/0.46 = ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ J ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % Nat.le_diff_conv2
% 0.19/0.46 thf(fact_148_le__diff__conv,axiom,
% 0.19/0.46 ! [J: nat,K: nat,I: nat] :
% 0.19/0.46 ( ( ord_less_eq_nat @ ( minus_minus_nat @ J @ K ) @ I )
% 0.19/0.46 = ( ord_less_eq_nat @ J @ ( plus_plus_nat @ I @ K ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % le_diff_conv
% 0.19/0.46 thf(fact_149_less__diff__conv2,axiom,
% 0.19/0.46 ! [K: nat,J: nat,I: nat] :
% 0.19/0.46 ( ( ord_less_eq_nat @ K @ J )
% 0.19/0.46 => ( ( ord_less_nat @ ( minus_minus_nat @ J @ K ) @ I )
% 0.19/0.46 = ( ord_less_nat @ J @ ( plus_plus_nat @ I @ K ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % less_diff_conv2
% 0.19/0.46 thf(fact_150_dual__order_Oantisym,axiom,
% 0.19/0.46 ! [B: nat,A: nat] :
% 0.19/0.46 ( ( ord_less_eq_nat @ B @ A )
% 0.19/0.46 => ( ( ord_less_eq_nat @ A @ B )
% 0.19/0.46 => ( A = B ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % dual_order.antisym
% 0.19/0.46 thf(fact_151_dual__order_Oeq__iff,axiom,
% 0.19/0.46 ( ( ^ [Y4: nat,Z: nat] : ( Y4 = Z ) )
% 0.19/0.46 = ( ^ [A3: nat,B3: nat] :
% 0.19/0.46 ( ( ord_less_eq_nat @ B3 @ A3 )
% 0.19/0.46 & ( ord_less_eq_nat @ A3 @ B3 ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % dual_order.eq_iff
% 0.19/0.46 thf(fact_152_dual__order_Otrans,axiom,
% 0.19/0.46 ! [B: nat,A: nat,C: nat] :
% 0.19/0.46 ( ( ord_less_eq_nat @ B @ A )
% 0.19/0.46 => ( ( ord_less_eq_nat @ C @ B )
% 0.19/0.46 => ( ord_less_eq_nat @ C @ A ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % dual_order.trans
% 0.19/0.46 thf(fact_153_linorder__wlog,axiom,
% 0.19/0.46 ! [P: nat > nat > $o,A: nat,B: nat] :
% 0.19/0.46 ( ! [A4: nat,B4: nat] :
% 0.19/0.46 ( ( ord_less_eq_nat @ A4 @ B4 )
% 0.19/0.46 => ( P @ A4 @ B4 ) )
% 0.19/0.46 => ( ! [A4: nat,B4: nat] :
% 0.19/0.46 ( ( P @ B4 @ A4 )
% 0.19/0.46 => ( P @ A4 @ B4 ) )
% 0.19/0.46 => ( P @ A @ B ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % linorder_wlog
% 0.19/0.46 thf(fact_154_dual__order_Orefl,axiom,
% 0.19/0.46 ! [A: nat] : ( ord_less_eq_nat @ A @ A ) ).
% 0.19/0.46
% 0.19/0.46 % dual_order.refl
% 0.19/0.46 thf(fact_155_order__trans,axiom,
% 0.19/0.46 ! [X: nat,Y: nat,Z3: nat] :
% 0.19/0.46 ( ( ord_less_eq_nat @ X @ Y )
% 0.19/0.46 => ( ( ord_less_eq_nat @ Y @ Z3 )
% 0.19/0.46 => ( ord_less_eq_nat @ X @ Z3 ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % order_trans
% 0.19/0.46 thf(fact_156_order__class_Oorder_Oantisym,axiom,
% 0.19/0.46 ! [A: nat,B: nat] :
% 0.19/0.46 ( ( ord_less_eq_nat @ A @ B )
% 0.19/0.46 => ( ( ord_less_eq_nat @ B @ A )
% 0.19/0.46 => ( A = B ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % order_class.order.antisym
% 0.19/0.46 thf(fact_157_ord__le__eq__trans,axiom,
% 0.19/0.46 ! [A: nat,B: nat,C: nat] :
% 0.19/0.46 ( ( ord_less_eq_nat @ A @ B )
% 0.19/0.46 => ( ( B = C )
% 0.19/0.46 => ( ord_less_eq_nat @ A @ C ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % ord_le_eq_trans
% 0.19/0.46 thf(fact_158_ord__eq__le__trans,axiom,
% 0.19/0.46 ! [A: nat,B: nat,C: nat] :
% 0.19/0.46 ( ( A = B )
% 0.19/0.46 => ( ( ord_less_eq_nat @ B @ C )
% 0.19/0.46 => ( ord_less_eq_nat @ A @ C ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % ord_eq_le_trans
% 0.19/0.46 thf(fact_159_order__class_Oorder_Oeq__iff,axiom,
% 0.19/0.46 ( ( ^ [Y4: nat,Z: nat] : ( Y4 = Z ) )
% 0.19/0.46 = ( ^ [A3: nat,B3: nat] :
% 0.19/0.46 ( ( ord_less_eq_nat @ A3 @ B3 )
% 0.19/0.46 & ( ord_less_eq_nat @ B3 @ A3 ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % order_class.order.eq_iff
% 0.19/0.46 thf(fact_160_antisym__conv,axiom,
% 0.19/0.46 ! [Y: nat,X: nat] :
% 0.19/0.46 ( ( ord_less_eq_nat @ Y @ X )
% 0.19/0.46 => ( ( ord_less_eq_nat @ X @ Y )
% 0.19/0.46 = ( X = Y ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % antisym_conv
% 0.19/0.46 thf(fact_161_le__cases3,axiom,
% 0.19/0.46 ! [X: nat,Y: nat,Z3: nat] :
% 0.19/0.46 ( ( ( ord_less_eq_nat @ X @ Y )
% 0.19/0.46 => ~ ( ord_less_eq_nat @ Y @ Z3 ) )
% 0.19/0.46 => ( ( ( ord_less_eq_nat @ Y @ X )
% 0.19/0.46 => ~ ( ord_less_eq_nat @ X @ Z3 ) )
% 0.19/0.46 => ( ( ( ord_less_eq_nat @ X @ Z3 )
% 0.19/0.46 => ~ ( ord_less_eq_nat @ Z3 @ Y ) )
% 0.19/0.46 => ( ( ( ord_less_eq_nat @ Z3 @ Y )
% 0.19/0.46 => ~ ( ord_less_eq_nat @ Y @ X ) )
% 0.19/0.46 => ( ( ( ord_less_eq_nat @ Y @ Z3 )
% 0.19/0.46 => ~ ( ord_less_eq_nat @ Z3 @ X ) )
% 0.19/0.46 => ~ ( ( ord_less_eq_nat @ Z3 @ X )
% 0.19/0.46 => ~ ( ord_less_eq_nat @ X @ Y ) ) ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % le_cases3
% 0.19/0.46 thf(fact_162_order_Otrans,axiom,
% 0.19/0.46 ! [A: nat,B: nat,C: nat] :
% 0.19/0.46 ( ( ord_less_eq_nat @ A @ B )
% 0.19/0.46 => ( ( ord_less_eq_nat @ B @ C )
% 0.19/0.46 => ( ord_less_eq_nat @ A @ C ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % order.trans
% 0.19/0.46 thf(fact_163_le__cases,axiom,
% 0.19/0.46 ! [X: nat,Y: nat] :
% 0.19/0.46 ( ~ ( ord_less_eq_nat @ X @ Y )
% 0.19/0.46 => ( ord_less_eq_nat @ Y @ X ) ) ).
% 0.19/0.46
% 0.19/0.46 % le_cases
% 0.19/0.46 thf(fact_164_eq__refl,axiom,
% 0.19/0.46 ! [X: nat,Y: nat] :
% 0.19/0.46 ( ( X = Y )
% 0.19/0.46 => ( ord_less_eq_nat @ X @ Y ) ) ).
% 0.19/0.46
% 0.19/0.46 % eq_refl
% 0.19/0.46 thf(fact_165_linear,axiom,
% 0.19/0.46 ! [X: nat,Y: nat] :
% 0.19/0.46 ( ( ord_less_eq_nat @ X @ Y )
% 0.19/0.46 | ( ord_less_eq_nat @ Y @ X ) ) ).
% 0.19/0.46
% 0.19/0.46 % linear
% 0.19/0.46 thf(fact_166_antisym,axiom,
% 0.19/0.46 ! [X: nat,Y: nat] :
% 0.19/0.46 ( ( ord_less_eq_nat @ X @ Y )
% 0.19/0.46 => ( ( ord_less_eq_nat @ Y @ X )
% 0.19/0.46 => ( X = Y ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % antisym
% 0.19/0.46 thf(fact_167_eq__iff,axiom,
% 0.19/0.46 ( ( ^ [Y4: nat,Z: nat] : ( Y4 = Z ) )
% 0.19/0.46 = ( ^ [X5: nat,Y5: nat] :
% 0.19/0.46 ( ( ord_less_eq_nat @ X5 @ Y5 )
% 0.19/0.46 & ( ord_less_eq_nat @ Y5 @ X5 ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % eq_iff
% 0.19/0.46 thf(fact_168_ord__le__eq__subst,axiom,
% 0.19/0.46 ! [A: nat,B: nat,F: nat > nat,C: nat] :
% 0.19/0.46 ( ( ord_less_eq_nat @ A @ B )
% 0.19/0.46 => ( ( ( F @ B )
% 0.19/0.46 = C )
% 0.19/0.46 => ( ! [X2: nat,Y2: nat] :
% 0.19/0.46 ( ( ord_less_eq_nat @ X2 @ Y2 )
% 0.19/0.46 => ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
% 0.19/0.46 => ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % ord_le_eq_subst
% 0.19/0.46 thf(fact_169_ord__eq__le__subst,axiom,
% 0.19/0.46 ! [A: nat,F: nat > nat,B: nat,C: nat] :
% 0.19/0.46 ( ( A
% 0.19/0.46 = ( F @ B ) )
% 0.19/0.46 => ( ( ord_less_eq_nat @ B @ C )
% 0.19/0.46 => ( ! [X2: nat,Y2: nat] :
% 0.19/0.46 ( ( ord_less_eq_nat @ X2 @ Y2 )
% 0.19/0.46 => ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
% 0.19/0.46 => ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % ord_eq_le_subst
% 0.19/0.46 thf(fact_170_order__subst2,axiom,
% 0.19/0.46 ! [A: nat,B: nat,F: nat > nat,C: nat] :
% 0.19/0.46 ( ( ord_less_eq_nat @ A @ B )
% 0.19/0.46 => ( ( ord_less_eq_nat @ ( F @ B ) @ C )
% 0.19/0.46 => ( ! [X2: nat,Y2: nat] :
% 0.19/0.46 ( ( ord_less_eq_nat @ X2 @ Y2 )
% 0.19/0.46 => ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
% 0.19/0.46 => ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % order_subst2
% 0.19/0.46 thf(fact_171_order__subst1,axiom,
% 0.19/0.46 ! [A: nat,F: nat > nat,B: nat,C: nat] :
% 0.19/0.46 ( ( ord_less_eq_nat @ A @ ( F @ B ) )
% 0.19/0.46 => ( ( ord_less_eq_nat @ B @ C )
% 0.19/0.46 => ( ! [X2: nat,Y2: nat] :
% 0.19/0.46 ( ( ord_less_eq_nat @ X2 @ Y2 )
% 0.19/0.46 => ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
% 0.19/0.46 => ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % order_subst1
% 0.19/0.46 thf(fact_172_dual__order_Ostrict__implies__not__eq,axiom,
% 0.19/0.46 ! [B: nat,A: nat] :
% 0.19/0.46 ( ( ord_less_nat @ B @ A )
% 0.19/0.46 => ( A != B ) ) ).
% 0.19/0.46
% 0.19/0.46 % dual_order.strict_implies_not_eq
% 0.19/0.46 thf(fact_173_order_Ostrict__implies__not__eq,axiom,
% 0.19/0.46 ! [A: nat,B: nat] :
% 0.19/0.46 ( ( ord_less_nat @ A @ B )
% 0.19/0.46 => ( A != B ) ) ).
% 0.19/0.46
% 0.19/0.46 % order.strict_implies_not_eq
% 0.19/0.46 thf(fact_174_not__less__iff__gr__or__eq,axiom,
% 0.19/0.46 ! [X: nat,Y: nat] :
% 0.19/0.46 ( ( ~ ( ord_less_nat @ X @ Y ) )
% 0.19/0.46 = ( ( ord_less_nat @ Y @ X )
% 0.19/0.46 | ( X = Y ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % not_less_iff_gr_or_eq
% 0.19/0.46 thf(fact_175_dual__order_Ostrict__trans,axiom,
% 0.19/0.46 ! [B: nat,A: nat,C: nat] :
% 0.19/0.46 ( ( ord_less_nat @ B @ A )
% 0.19/0.46 => ( ( ord_less_nat @ C @ B )
% 0.19/0.46 => ( ord_less_nat @ C @ A ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % dual_order.strict_trans
% 0.19/0.46 thf(fact_176_linorder__less__wlog,axiom,
% 0.19/0.46 ! [P: nat > nat > $o,A: nat,B: nat] :
% 0.19/0.46 ( ! [A4: nat,B4: nat] :
% 0.19/0.46 ( ( ord_less_nat @ A4 @ B4 )
% 0.19/0.46 => ( P @ A4 @ B4 ) )
% 0.19/0.46 => ( ! [A4: nat] : ( P @ A4 @ A4 )
% 0.19/0.46 => ( ! [A4: nat,B4: nat] :
% 0.19/0.46 ( ( P @ B4 @ A4 )
% 0.19/0.46 => ( P @ A4 @ B4 ) )
% 0.19/0.46 => ( P @ A @ B ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % linorder_less_wlog
% 0.19/0.46 thf(fact_177_exists__least__iff,axiom,
% 0.19/0.46 ( ( ^ [P2: nat > $o] :
% 0.19/0.46 ? [X3: nat] : ( P2 @ X3 ) )
% 0.19/0.46 = ( ^ [P3: nat > $o] :
% 0.19/0.46 ? [N2: nat] :
% 0.19/0.46 ( ( P3 @ N2 )
% 0.19/0.46 & ! [M2: nat] :
% 0.19/0.46 ( ( ord_less_nat @ M2 @ N2 )
% 0.19/0.46 => ~ ( P3 @ M2 ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % exists_least_iff
% 0.19/0.46 thf(fact_178_less__imp__not__less,axiom,
% 0.19/0.46 ! [X: nat,Y: nat] :
% 0.19/0.46 ( ( ord_less_nat @ X @ Y )
% 0.19/0.46 => ~ ( ord_less_nat @ Y @ X ) ) ).
% 0.19/0.46
% 0.19/0.46 % less_imp_not_less
% 0.19/0.46 thf(fact_179_order_Ostrict__trans,axiom,
% 0.19/0.46 ! [A: nat,B: nat,C: nat] :
% 0.19/0.46 ( ( ord_less_nat @ A @ B )
% 0.19/0.46 => ( ( ord_less_nat @ B @ C )
% 0.19/0.46 => ( ord_less_nat @ A @ C ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % order.strict_trans
% 0.19/0.46 thf(fact_180_dual__order_Oirrefl,axiom,
% 0.19/0.46 ! [A: nat] :
% 0.19/0.46 ~ ( ord_less_nat @ A @ A ) ).
% 0.19/0.46
% 0.19/0.46 % dual_order.irrefl
% 0.19/0.46 thf(fact_181_linorder__cases,axiom,
% 0.19/0.46 ! [X: nat,Y: nat] :
% 0.19/0.46 ( ~ ( ord_less_nat @ X @ Y )
% 0.19/0.46 => ( ( X != Y )
% 0.19/0.46 => ( ord_less_nat @ Y @ X ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % linorder_cases
% 0.19/0.46 thf(fact_182_less__imp__triv,axiom,
% 0.19/0.46 ! [X: nat,Y: nat,P: $o] :
% 0.19/0.46 ( ( ord_less_nat @ X @ Y )
% 0.19/0.46 => ( ( ord_less_nat @ Y @ X )
% 0.19/0.46 => P ) ) ).
% 0.19/0.46
% 0.19/0.46 % less_imp_triv
% 0.19/0.46 thf(fact_183_less__imp__not__eq2,axiom,
% 0.19/0.46 ! [X: nat,Y: nat] :
% 0.19/0.46 ( ( ord_less_nat @ X @ Y )
% 0.19/0.46 => ( Y != X ) ) ).
% 0.19/0.46
% 0.19/0.46 % less_imp_not_eq2
% 0.19/0.46 thf(fact_184_antisym__conv3,axiom,
% 0.19/0.46 ! [Y: nat,X: nat] :
% 0.19/0.46 ( ~ ( ord_less_nat @ Y @ X )
% 0.19/0.46 => ( ( ~ ( ord_less_nat @ X @ Y ) )
% 0.19/0.46 = ( X = Y ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % antisym_conv3
% 0.19/0.46 thf(fact_185_less__induct,axiom,
% 0.19/0.46 ! [P: nat > $o,A: nat] :
% 0.19/0.46 ( ! [X2: nat] :
% 0.19/0.46 ( ! [Y3: nat] :
% 0.19/0.46 ( ( ord_less_nat @ Y3 @ X2 )
% 0.19/0.46 => ( P @ Y3 ) )
% 0.19/0.46 => ( P @ X2 ) )
% 0.19/0.46 => ( P @ A ) ) ).
% 0.19/0.46
% 0.19/0.46 % less_induct
% 0.19/0.46 thf(fact_186_less__not__sym,axiom,
% 0.19/0.46 ! [X: nat,Y: nat] :
% 0.19/0.46 ( ( ord_less_nat @ X @ Y )
% 0.19/0.46 => ~ ( ord_less_nat @ Y @ X ) ) ).
% 0.19/0.46
% 0.19/0.46 % less_not_sym
% 0.19/0.46 thf(fact_187_less__imp__not__eq,axiom,
% 0.19/0.46 ! [X: nat,Y: nat] :
% 0.19/0.46 ( ( ord_less_nat @ X @ Y )
% 0.19/0.46 => ( X != Y ) ) ).
% 0.19/0.46
% 0.19/0.46 % less_imp_not_eq
% 0.19/0.46 thf(fact_188_dual__order_Oasym,axiom,
% 0.19/0.46 ! [B: nat,A: nat] :
% 0.19/0.46 ( ( ord_less_nat @ B @ A )
% 0.19/0.46 => ~ ( ord_less_nat @ A @ B ) ) ).
% 0.19/0.46
% 0.19/0.46 % dual_order.asym
% 0.19/0.46 thf(fact_189_ord__less__eq__trans,axiom,
% 0.19/0.46 ! [A: nat,B: nat,C: nat] :
% 0.19/0.46 ( ( ord_less_nat @ A @ B )
% 0.19/0.46 => ( ( B = C )
% 0.19/0.46 => ( ord_less_nat @ A @ C ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % ord_less_eq_trans
% 0.19/0.46 thf(fact_190_ord__eq__less__trans,axiom,
% 0.19/0.46 ! [A: nat,B: nat,C: nat] :
% 0.19/0.46 ( ( A = B )
% 0.19/0.46 => ( ( ord_less_nat @ B @ C )
% 0.19/0.46 => ( ord_less_nat @ A @ C ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % ord_eq_less_trans
% 0.19/0.46 thf(fact_191_less__irrefl,axiom,
% 0.19/0.46 ! [X: nat] :
% 0.19/0.46 ~ ( ord_less_nat @ X @ X ) ).
% 0.19/0.46
% 0.19/0.46 % less_irrefl
% 0.19/0.46 thf(fact_192_less__linear,axiom,
% 0.19/0.46 ! [X: nat,Y: nat] :
% 0.19/0.46 ( ( ord_less_nat @ X @ Y )
% 0.19/0.46 | ( X = Y )
% 0.19/0.46 | ( ord_less_nat @ Y @ X ) ) ).
% 0.19/0.46
% 0.19/0.46 % less_linear
% 0.19/0.46 thf(fact_193_less__trans,axiom,
% 0.19/0.46 ! [X: nat,Y: nat,Z3: nat] :
% 0.19/0.46 ( ( ord_less_nat @ X @ Y )
% 0.19/0.46 => ( ( ord_less_nat @ Y @ Z3 )
% 0.19/0.46 => ( ord_less_nat @ X @ Z3 ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % less_trans
% 0.19/0.46 thf(fact_194_less__asym_H,axiom,
% 0.19/0.46 ! [A: nat,B: nat] :
% 0.19/0.46 ( ( ord_less_nat @ A @ B )
% 0.19/0.46 => ~ ( ord_less_nat @ B @ A ) ) ).
% 0.19/0.46
% 0.19/0.46 % less_asym'
% 0.19/0.46 thf(fact_195_less__asym,axiom,
% 0.19/0.46 ! [X: nat,Y: nat] :
% 0.19/0.46 ( ( ord_less_nat @ X @ Y )
% 0.19/0.46 => ~ ( ord_less_nat @ Y @ X ) ) ).
% 0.19/0.46
% 0.19/0.46 % less_asym
% 0.19/0.46 thf(fact_196_less__imp__neq,axiom,
% 0.19/0.46 ! [X: nat,Y: nat] :
% 0.19/0.46 ( ( ord_less_nat @ X @ Y )
% 0.19/0.46 => ( X != Y ) ) ).
% 0.19/0.46
% 0.19/0.46 % less_imp_neq
% 0.19/0.46 thf(fact_197_order_Oasym,axiom,
% 0.19/0.46 ! [A: nat,B: nat] :
% 0.19/0.46 ( ( ord_less_nat @ A @ B )
% 0.19/0.46 => ~ ( ord_less_nat @ B @ A ) ) ).
% 0.19/0.46
% 0.19/0.46 % order.asym
% 0.19/0.46 thf(fact_198_neq__iff,axiom,
% 0.19/0.46 ! [X: nat,Y: nat] :
% 0.19/0.46 ( ( X != Y )
% 0.19/0.46 = ( ( ord_less_nat @ X @ Y )
% 0.19/0.46 | ( ord_less_nat @ Y @ X ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % neq_iff
% 0.19/0.46 thf(fact_199_neqE,axiom,
% 0.19/0.46 ! [X: nat,Y: nat] :
% 0.19/0.46 ( ( X != Y )
% 0.19/0.46 => ( ~ ( ord_less_nat @ X @ Y )
% 0.19/0.46 => ( ord_less_nat @ Y @ X ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % neqE
% 0.19/0.46 thf(fact_200_gt__ex,axiom,
% 0.19/0.46 ! [X: nat] :
% 0.19/0.46 ? [X_1: nat] : ( ord_less_nat @ X @ X_1 ) ).
% 0.19/0.46
% 0.19/0.46 % gt_ex
% 0.19/0.46 thf(fact_201_order__less__subst2,axiom,
% 0.19/0.46 ! [A: nat,B: nat,F: nat > nat,C: nat] :
% 0.19/0.46 ( ( ord_less_nat @ A @ B )
% 0.19/0.46 => ( ( ord_less_nat @ ( F @ B ) @ C )
% 0.19/0.46 => ( ! [X2: nat,Y2: nat] :
% 0.19/0.46 ( ( ord_less_nat @ X2 @ Y2 )
% 0.19/0.46 => ( ord_less_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
% 0.19/0.46 => ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % order_less_subst2
% 0.19/0.46 thf(fact_202_order__less__subst1,axiom,
% 0.19/0.46 ! [A: nat,F: nat > nat,B: nat,C: nat] :
% 0.19/0.46 ( ( ord_less_nat @ A @ ( F @ B ) )
% 0.19/0.46 => ( ( ord_less_nat @ B @ C )
% 0.19/0.46 => ( ! [X2: nat,Y2: nat] :
% 0.19/0.46 ( ( ord_less_nat @ X2 @ Y2 )
% 0.19/0.46 => ( ord_less_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
% 0.19/0.46 => ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % order_less_subst1
% 0.19/0.46 thf(fact_203_ord__less__eq__subst,axiom,
% 0.19/0.46 ! [A: nat,B: nat,F: nat > nat,C: nat] :
% 0.19/0.46 ( ( ord_less_nat @ A @ B )
% 0.19/0.46 => ( ( ( F @ B )
% 0.19/0.46 = C )
% 0.19/0.46 => ( ! [X2: nat,Y2: nat] :
% 0.19/0.46 ( ( ord_less_nat @ X2 @ Y2 )
% 0.19/0.46 => ( ord_less_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
% 0.19/0.46 => ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % ord_less_eq_subst
% 0.19/0.46 thf(fact_204_ord__eq__less__subst,axiom,
% 0.19/0.46 ! [A: nat,F: nat > nat,B: nat,C: nat] :
% 0.19/0.46 ( ( A
% 0.19/0.46 = ( F @ B ) )
% 0.19/0.46 => ( ( ord_less_nat @ B @ C )
% 0.19/0.46 => ( ! [X2: nat,Y2: nat] :
% 0.19/0.46 ( ( ord_less_nat @ X2 @ Y2 )
% 0.19/0.46 => ( ord_less_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
% 0.19/0.46 => ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % ord_eq_less_subst
% 0.19/0.46 thf(fact_205_pinf_I1_J,axiom,
% 0.19/0.46 ! [P: nat > $o,P4: nat > $o,Q: nat > $o,Q2: nat > $o] :
% 0.19/0.46 ( ? [Z4: nat] :
% 0.19/0.46 ! [X2: nat] :
% 0.19/0.46 ( ( ord_less_nat @ Z4 @ X2 )
% 0.19/0.46 => ( ( P @ X2 )
% 0.19/0.46 = ( P4 @ X2 ) ) )
% 0.19/0.46 => ( ? [Z4: nat] :
% 0.19/0.46 ! [X2: nat] :
% 0.19/0.46 ( ( ord_less_nat @ Z4 @ X2 )
% 0.19/0.46 => ( ( Q @ X2 )
% 0.19/0.46 = ( Q2 @ X2 ) ) )
% 0.19/0.46 => ? [Z2: nat] :
% 0.19/0.46 ! [X4: nat] :
% 0.19/0.46 ( ( ord_less_nat @ Z2 @ X4 )
% 0.19/0.46 => ( ( ( P @ X4 )
% 0.19/0.46 & ( Q @ X4 ) )
% 0.19/0.46 = ( ( P4 @ X4 )
% 0.19/0.46 & ( Q2 @ X4 ) ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % pinf(1)
% 0.19/0.46 thf(fact_206_pinf_I2_J,axiom,
% 0.19/0.46 ! [P: nat > $o,P4: nat > $o,Q: nat > $o,Q2: nat > $o] :
% 0.19/0.46 ( ? [Z4: nat] :
% 0.19/0.46 ! [X2: nat] :
% 0.19/0.46 ( ( ord_less_nat @ Z4 @ X2 )
% 0.19/0.46 => ( ( P @ X2 )
% 0.19/0.46 = ( P4 @ X2 ) ) )
% 0.19/0.46 => ( ? [Z4: nat] :
% 0.19/0.46 ! [X2: nat] :
% 0.19/0.46 ( ( ord_less_nat @ Z4 @ X2 )
% 0.19/0.46 => ( ( Q @ X2 )
% 0.19/0.46 = ( Q2 @ X2 ) ) )
% 0.19/0.46 => ? [Z2: nat] :
% 0.19/0.46 ! [X4: nat] :
% 0.19/0.46 ( ( ord_less_nat @ Z2 @ X4 )
% 0.19/0.46 => ( ( ( P @ X4 )
% 0.19/0.46 | ( Q @ X4 ) )
% 0.19/0.46 = ( ( P4 @ X4 )
% 0.19/0.46 | ( Q2 @ X4 ) ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % pinf(2)
% 0.19/0.46 thf(fact_207_pinf_I3_J,axiom,
% 0.19/0.46 ! [T: nat] :
% 0.19/0.46 ? [Z2: nat] :
% 0.19/0.46 ! [X4: nat] :
% 0.19/0.46 ( ( ord_less_nat @ Z2 @ X4 )
% 0.19/0.46 => ( X4 != T ) ) ).
% 0.19/0.46
% 0.19/0.46 % pinf(3)
% 0.19/0.46 thf(fact_208_pinf_I4_J,axiom,
% 0.19/0.46 ! [T: nat] :
% 0.19/0.46 ? [Z2: nat] :
% 0.19/0.46 ! [X4: nat] :
% 0.19/0.46 ( ( ord_less_nat @ Z2 @ X4 )
% 0.19/0.46 => ( X4 != T ) ) ).
% 0.19/0.46
% 0.19/0.46 % pinf(4)
% 0.19/0.46 thf(fact_209_pinf_I5_J,axiom,
% 0.19/0.46 ! [T: nat] :
% 0.19/0.46 ? [Z2: nat] :
% 0.19/0.46 ! [X4: nat] :
% 0.19/0.46 ( ( ord_less_nat @ Z2 @ X4 )
% 0.19/0.46 => ~ ( ord_less_nat @ X4 @ T ) ) ).
% 0.19/0.46
% 0.19/0.46 % pinf(5)
% 0.19/0.46 thf(fact_210_pinf_I7_J,axiom,
% 0.19/0.46 ! [T: nat] :
% 0.19/0.46 ? [Z2: nat] :
% 0.19/0.46 ! [X4: nat] :
% 0.19/0.46 ( ( ord_less_nat @ Z2 @ X4 )
% 0.19/0.46 => ( ord_less_nat @ T @ X4 ) ) ).
% 0.19/0.46
% 0.19/0.46 % pinf(7)
% 0.19/0.46 thf(fact_211_minf_I1_J,axiom,
% 0.19/0.46 ! [P: nat > $o,P4: nat > $o,Q: nat > $o,Q2: nat > $o] :
% 0.19/0.46 ( ? [Z4: nat] :
% 0.19/0.46 ! [X2: nat] :
% 0.19/0.46 ( ( ord_less_nat @ X2 @ Z4 )
% 0.19/0.46 => ( ( P @ X2 )
% 0.19/0.46 = ( P4 @ X2 ) ) )
% 0.19/0.46 => ( ? [Z4: nat] :
% 0.19/0.46 ! [X2: nat] :
% 0.19/0.46 ( ( ord_less_nat @ X2 @ Z4 )
% 0.19/0.46 => ( ( Q @ X2 )
% 0.19/0.46 = ( Q2 @ X2 ) ) )
% 0.19/0.46 => ? [Z2: nat] :
% 0.19/0.46 ! [X4: nat] :
% 0.19/0.46 ( ( ord_less_nat @ X4 @ Z2 )
% 0.19/0.46 => ( ( ( P @ X4 )
% 0.19/0.46 & ( Q @ X4 ) )
% 0.19/0.46 = ( ( P4 @ X4 )
% 0.19/0.46 & ( Q2 @ X4 ) ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % minf(1)
% 0.19/0.46 thf(fact_212_minf_I2_J,axiom,
% 0.19/0.46 ! [P: nat > $o,P4: nat > $o,Q: nat > $o,Q2: nat > $o] :
% 0.19/0.46 ( ? [Z4: nat] :
% 0.19/0.46 ! [X2: nat] :
% 0.19/0.46 ( ( ord_less_nat @ X2 @ Z4 )
% 0.19/0.46 => ( ( P @ X2 )
% 0.19/0.46 = ( P4 @ X2 ) ) )
% 0.19/0.46 => ( ? [Z4: nat] :
% 0.19/0.46 ! [X2: nat] :
% 0.19/0.46 ( ( ord_less_nat @ X2 @ Z4 )
% 0.19/0.46 => ( ( Q @ X2 )
% 0.19/0.46 = ( Q2 @ X2 ) ) )
% 0.19/0.46 => ? [Z2: nat] :
% 0.19/0.46 ! [X4: nat] :
% 0.19/0.46 ( ( ord_less_nat @ X4 @ Z2 )
% 0.19/0.46 => ( ( ( P @ X4 )
% 0.19/0.46 | ( Q @ X4 ) )
% 0.19/0.46 = ( ( P4 @ X4 )
% 0.19/0.46 | ( Q2 @ X4 ) ) ) ) ) ) ).
% 0.19/0.46
% 0.19/0.46 % minf(2)
% 0.19/0.46 thf(fact_213_minf_I3_J,axiom,
% 0.19/0.46 ! [T: nat] :
% 0.19/0.46 ? [Z2: nat] :
% 0.19/0.46 ! [X4: nat] :
% 0.19/0.46 ( ( ord_less_nat @ X4 @ Z2 )
% 0.19/0.46 => ( X4 != T ) ) ).
% 0.19/0.46
% 0.19/0.46 % minf(3)
% 0.19/0.46
% 0.19/0.46 % Conjectures (1)
% 0.19/0.46 thf(conj_0,conjecture,
% 0.19/0.46 ( ( last_c571238084t_unit @ ( fe @ n ) )
% 0.19/0.46 = ( nth_co1649820636t_unit @ ( fe @ n ) @ ( minus_minus_nat @ ( size_s1406904903t_unit @ ( fe @ n ) ) @ one_one_nat ) ) ) ).
% 0.19/0.46
% 0.19/0.46 %------------------------------------------------------------------------------
% 0.19/0.46 ------------------------
% 0.30/1.80 ---- Embedded in THF ---
% 0.30/1.80 %%% This output was generated by embedproblem, version 1.9.5 (library version 1.9).
% 0.30/1.80 %%% Generated on Tue Feb 24 08:24:55 EST 2026
% 0.30/1.80 %%% using '$$dhol' embedding, version 1.3.0.
% 0.30/1.80 %%% Logic specification used:
% 0.30/1.80 %%% thf(spec, logic, $$dhol).
% 0.30/1.80
% 0.30/1.80 % SZS output start ListOfTHF for /export/starexec/sandbox/tmp/tmp.iTgpky4eW3/DTF2DTF32211.p
% See solution above
% 0.30/1.80 ------------------------
% 0.30/1.82 ---- Cleaned THF ---
% 0.30/1.82 thf(ty_n_t__List__Olist_It__AsynchronousSystem__Oconfiguration__Oconfiguration____ext_Itf__p_Mtf__v_Mtf__s_Mt__Product____Type__Ounit_J_J,type,
% 0.30/1.82 list_c1059388851t_unit: $tType ).
% 0.30/1.82
% 0.30/1.82 thf(ty_n_t__AsynchronousSystem__Oconfiguration__Oconfiguration____ext_Itf__p_Mtf__v_Mtf__s_Mt__Product____Type__Ounit_J,type,
% 0.30/1.82 config256849571t_unit: $tType ).
% 0.30/1.82
% 0.30/1.82 thf(ty_n_t__Nat__Onat,type,
% 0.30/1.82 nat: $tType ).
% 0.30/1.82
% 0.30/1.82 thf(sy_c_Groups_Ominus__class_Ominus_001t__Nat__Onat,type,
% 0.30/1.82 minus_minus_nat: nat > nat > nat ).
% 0.30/1.82
% 0.30/1.82 thf(sy_c_Groups_Oone__class_Oone_001t__Nat__Onat,type,
% 0.30/1.82 one_one_nat: nat ).
% 0.30/1.82
% 0.30/1.82 thf(sy_c_Groups_Oplus__class_Oplus_001t__Nat__Onat,type,
% 0.30/1.82 plus_plus_nat: nat > nat > nat ).
% 0.30/1.82
% 0.30/1.82 thf(sy_c_ListUtilities_OprefixList_001t__AsynchronousSystem__Oconfiguration__Oconfiguration____ext_Itf__p_Mtf__v_Mtf__s_Mt__Product____Type__Ounit_J,type,
% 0.30/1.82 prefix1615116500t_unit: list_c1059388851t_unit > list_c1059388851t_unit > $o ).
% 0.30/1.82
% 0.30/1.82 thf(sy_c_List_Olast_001t__AsynchronousSystem__Oconfiguration__Oconfiguration____ext_Itf__p_Mtf__v_Mtf__s_Mt__Product____Type__Ounit_J,type,
% 0.30/1.82 last_c571238084t_unit: list_c1059388851t_unit > config256849571t_unit ).
% 0.30/1.82
% 0.30/1.82 thf(sy_c_List_Olist_ONil_001t__AsynchronousSystem__Oconfiguration__Oconfiguration____ext_Itf__p_Mtf__v_Mtf__s_Mt__Product____Type__Ounit_J,type,
% 0.30/1.82 nil_co1338500125t_unit: list_c1059388851t_unit ).
% 0.30/1.82
% 0.30/1.82 thf(sy_c_List_Onth_001t__AsynchronousSystem__Oconfiguration__Oconfiguration____ext_Itf__p_Mtf__v_Mtf__s_Mt__Product____Type__Ounit_J,type,
% 0.30/1.82 nth_co1649820636t_unit: list_c1059388851t_unit > nat > config256849571t_unit ).
% 0.30/1.82
% 0.30/1.82 thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__AsynchronousSystem__Oconfiguration__Oconfiguration____ext_Itf__p_Mtf__v_Mtf__s_Mt__Product____Type__Ounit_J_J,type,
% 0.30/1.82 size_s1406904903t_unit: list_c1059388851t_unit > nat ).
% 0.30/1.82
% 0.30/1.82 thf(sy_c_Orderings_Oord__class_Oless_001t__Nat__Onat,type,
% 0.30/1.82 ord_less_nat: nat > nat > $o ).
% 0.30/1.82
% 0.30/1.82 thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Nat__Onat,type,
% 0.30/1.82 ord_less_eq_nat: nat > nat > $o ).
% 0.30/1.82
% 0.30/1.82 thf(sy_v_fe____,type,
% 0.30/1.82 fe: nat > list_c1059388851t_unit ).
% 0.30/1.82
% 0.30/1.82 thf(sy_v_n0____,type,
% 0.30/1.82 n0: nat ).
% 0.30/1.82
% 0.30/1.82 thf(sy_v_n____,type,
% 0.30/1.82 n: nat ).
% 0.30/1.82
% 0.30/1.82 thf(fact_0_FeNNotEmpty,axiom,
% 0.30/1.82 ( ( fe @ n )
% 0.30/1.82 != nil_co1338500125t_unit ) ).
% 0.30/1.82
% 0.30/1.82 thf(fact_1_LastIsLastIndex,axiom,
% 0.30/1.82 ! [L: list_c1059388851t_unit] :
% 0.30/1.82 ( ( L != nil_co1338500125t_unit )
% 0.30/1.82 => ( ( last_c571238084t_unit @ L )
% 0.30/1.82 = ( nth_co1649820636t_unit @ L @ ( minus_minus_nat @ ( size_s1406904903t_unit @ L ) @ one_one_nat ) ) ) ) ).
% 0.30/1.82
% 0.30/1.82 thf(fact_2_Fou2,axiom,
% 0.30/1.82 ord_less_nat @ ( minus_minus_nat @ ( size_s1406904903t_unit @ ( fe @ n ) ) @ one_one_nat ) @ ( size_s1406904903t_unit @ ( fe @ n ) ) ).
% 0.30/1.82
% 0.30/1.82 thf(fact_3_AssmNoDecided_I1_J,axiom,
% 0.30/1.82 ord_less_nat @ n0 @ ( size_s1406904903t_unit @ ( fe @ n ) ) ).
% 0.30/1.82
% 0.30/1.82 thf(fact_4_last__conv__nth,axiom,
% 0.30/1.82 ! [Xs: list_c1059388851t_unit] :
% 0.30/1.82 ( ( Xs != nil_co1338500125t_unit )
% 0.30/1.82 => ( ( last_c571238084t_unit @ Xs )
% 0.30/1.82 = ( nth_co1649820636t_unit @ Xs @ ( minus_minus_nat @ ( size_s1406904903t_unit @ Xs ) @ one_one_nat ) ) ) ) ).
% 0.30/1.82
% 0.30/1.82 thf(fact_5_Fou,axiom,
% 0.30/1.82 ord_less_eq_nat @ n0 @ ( minus_minus_nat @ ( size_s1406904903t_unit @ ( fe @ n ) ) @ one_one_nat ) ).
% 0.30/1.82
% 0.30/1.82 thf(fact_6_one__natural_Orsp,axiom,
% 0.30/1.82 one_one_nat = one_one_nat ).
% 0.30/1.82
% 0.30/1.82 thf(fact_7_diff__commute,axiom,
% 0.30/1.82 ! [I: nat,J: nat,K: nat] :
% 0.30/1.82 ( ( minus_minus_nat @ ( minus_minus_nat @ I @ J ) @ K )
% 0.30/1.82 = ( minus_minus_nat @ ( minus_minus_nat @ I @ K ) @ J ) ) ).
% 0.30/1.82
% 0.30/1.82 thf(fact_8_Ex__list__of__length,axiom,
% 0.30/1.82 ! [N: nat] :
% 0.30/1.82 ? [Xs2: list_c1059388851t_unit] :
% 0.30/1.82 ( ( size_s1406904903t_unit @ Xs2 )
% 0.30/1.82 = N ) ).
% 0.30/1.82
% 0.30/1.82 thf(fact_9_neq__if__length__neq,axiom,
% 0.30/1.82 ! [Xs: list_c1059388851t_unit,Ys: list_c1059388851t_unit] :
% 0.30/1.82 ( ( ( size_s1406904903t_unit @ Xs )
% 0.30/1.82 != ( size_s1406904903t_unit @ Ys ) )
% 0.30/1.82 => ( Xs != Ys ) ) ).
% 0.30/1.82
% 0.30/1.82 thf(fact_10_size__neq__size__imp__neq,axiom,
% 0.30/1.82 ! [X: list_c1059388851t_unit,Y: list_c1059388851t_unit] :
% 0.30/1.82 ( ( ( size_s1406904903t_unit @ X )
% 0.30/1.82 != ( size_s1406904903t_unit @ Y ) )
% 0.30/1.82 => ( X != Y ) ) ).
% 0.30/1.82
% 0.30/1.82 thf(fact_11_diff__diff__cancel,axiom,
% 0.30/1.82 ! [I: nat,N: nat] :
% 0.30/1.82 ( ( ord_less_eq_nat @ I @ N )
% 0.30/1.82 => ( ( minus_minus_nat @ N @ ( minus_minus_nat @ N @ I ) )
% 0.30/1.82 = I ) ) ).
% 0.30/1.82
% 0.30/1.82 thf(fact_12_le__refl,axiom,
% 0.30/1.82 ! [N: nat] : ( ord_less_eq_nat @ N @ N ) ).
% 0.30/1.82
% 0.30/1.82 thf(fact_13_le__trans,axiom,
% 0.30/1.82 ! [I: nat,J: nat,K: nat] :
% 0.30/1.82 ( ( ord_less_eq_nat @ I @ J )
% 0.30/1.82 => ( ( ord_less_eq_nat @ J @ K )
% 0.30/1.82 => ( ord_less_eq_nat @ I @ K ) ) ) ).
% 0.30/1.82
% 0.30/1.82 thf(fact_14_eq__imp__le,axiom,
% 0.30/1.82 ! [M: nat,N: nat] :
% 0.30/1.82 ( ( M = N )
% 0.30/1.82 => ( ord_less_eq_nat @ M @ N ) ) ).
% 0.30/1.82
% 0.30/1.82 thf(fact_15_le__antisym,axiom,
% 0.30/1.82 ! [M: nat,N: nat] :
% 0.30/1.82 ( ( ord_less_eq_nat @ M @ N )
% 0.30/1.82 => ( ( ord_less_eq_nat @ N @ M )
% 0.30/1.82 => ( M = N ) ) ) ).
% 0.30/1.82
% 0.30/1.82 thf(fact_16_nat__less__le,axiom,
% 0.30/1.82 ( ord_less_nat
% 0.30/1.82 = ( ^ [M2: nat,N2: nat] :
% 0.30/1.82 ( ( ord_less_eq_nat @ M2 @ N2 )
% 0.30/1.82 & ( M2 != N2 ) ) ) ) ).
% 0.30/1.82
% 0.30/1.82 thf(fact_17_nat__neq__iff,axiom,
% 0.30/1.82 ! [M: nat,N: nat] :
% 0.30/1.82 ( ( M != N )
% 0.30/1.82 = ( ( ord_less_nat @ M @ N )
% 0.30/1.82 | ( ord_less_nat @ N @ M ) ) ) ).
% 0.30/1.82
% 0.30/1.82 thf(fact_18_less__diff__iff,axiom,
% 0.30/1.82 ! [K: nat,M: nat,N: nat] :
% 0.30/1.82 ( ( ord_less_eq_nat @ K @ M )
% 0.30/1.82 => ( ( ord_less_eq_nat @ K @ N )
% 0.30/1.82 => ( ( ord_less_nat @ ( minus_minus_nat @ M @ K ) @ ( minus_minus_nat @ N @ K ) )
% 0.30/1.82 = ( ord_less_nat @ M @ N ) ) ) ) ).
% 0.30/1.82
% 0.30/1.82 thf(fact_19_less__not__refl,axiom,
% 0.30/1.82 ! [N: nat] :
% 0.30/1.82 ~ ( ord_less_nat @ N @ N ) ).
% 0.30/1.82
% 0.30/1.82 thf(fact_20_nat__le__linear,axiom,
% 0.30/1.82 ! [M: nat,N: nat] :
% 0.30/1.82 ( ( ord_less_eq_nat @ M @ N )
% 0.30/1.82 | ( ord_less_eq_nat @ N @ M ) ) ).
% 0.30/1.82
% 0.30/1.82 thf(fact_21_diff__less__mono,axiom,
% 0.30/1.82 ! [A: nat,B: nat,C: nat] :
% 0.30/1.82 ( ( ord_less_nat @ A @ B )
% 0.30/1.82 => ( ( ord_less_eq_nat @ C @ A )
% 0.30/1.82 => ( ord_less_nat @ ( minus_minus_nat @ A @ C ) @ ( minus_minus_nat @ B @ C ) ) ) ) ).
% 0.30/1.82
% 0.30/1.82 thf(fact_22_less__not__refl2,axiom,
% 0.30/1.82 ! [N: nat,M: nat] :
% 0.30/1.82 ( ( ord_less_nat @ N @ M )
% 0.30/1.82 => ( M != N ) ) ).
% 0.30/1.82
% 0.30/1.82 thf(fact_23_less__not__refl3,axiom,
% 0.30/1.82 ! [S: nat,T: nat] :
% 0.30/1.82 ( ( ord_less_nat @ S @ T )
% 0.30/1.82 => ( S != T ) ) ).
% 0.30/1.82
% 0.30/1.82 thf(fact_24_less__imp__le__nat,axiom,
% 0.30/1.82 ! [M: nat,N: nat] :
% 0.30/1.82 ( ( ord_less_nat @ M @ N )
% 0.30/1.82 => ( ord_less_eq_nat @ M @ N ) ) ).
% 0.30/1.82
% 0.30/1.82 thf(fact_25_less__irrefl__nat,axiom,
% 0.30/1.82 ! [N: nat] :
% 0.30/1.82 ~ ( ord_less_nat @ N @ N ) ).
% 0.30/1.82
% 0.30/1.82 thf(fact_26_nat__less__induct,axiom,
% 0.30/1.82 ! [P: nat > $o,N: nat] :
% 0.30/1.82 ( ! [N3: nat] :
% 0.30/1.82 ( ! [M3: nat] :
% 0.30/1.82 ( ( ord_less_nat @ M3 @ N3 )
% 0.30/1.82 => ( P @ M3 ) )
% 0.30/1.82 => ( P @ N3 ) )
% 0.30/1.82 => ( P @ N ) ) ).
% 0.30/1.82
% 0.30/1.82 thf(fact_27_infinite__descent,axiom,
% 0.30/1.82 ! [P: nat > $o,N: nat] :
% 0.30/1.82 ( ! [N3: nat] :
% 0.30/1.82 ( ~ ( P @ N3 )
% 0.30/1.82 => ? [M3: nat] :
% 0.30/1.82 ( ( ord_less_nat @ M3 @ N3 )
% 0.30/1.82 & ~ ( P @ M3 ) ) )
% 0.30/1.82 => ( P @ N ) ) ).
% 0.30/1.82
% 0.30/1.82 thf(fact_28_le__eq__less__or__eq,axiom,
% 0.30/1.82 ( ord_less_eq_nat
% 0.30/1.82 = ( ^ [M2: nat,N2: nat] :
% 0.30/1.82 ( ( ord_less_nat @ M2 @ N2 )
% 0.30/1.82 | ( M2 = N2 ) ) ) ) ).
% 0.30/1.82
% 0.30/1.82 thf(fact_29_less__or__eq__imp__le,axiom,
% 0.30/1.82 ! [M: nat,N: nat] :
% 0.30/1.82 ( ( ( ord_less_nat @ M @ N )
% 0.30/1.82 | ( M = N ) )
% 0.30/1.82 => ( ord_less_eq_nat @ M @ N ) ) ).
% 0.30/1.82
% 0.30/1.82 thf(fact_30_linorder__neqE__nat,axiom,
% 0.30/1.82 ! [X: nat,Y: nat] :
% 0.30/1.82 ( ( X != Y )
% 0.30/1.82 => ( ~ ( ord_less_nat @ X @ Y )
% 0.30/1.82 => ( ord_less_nat @ Y @ X ) ) ) ).
% 0.30/1.82
% 0.30/1.82 thf(fact_31_Nat_Oex__has__greatest__nat,axiom,
% 0.30/1.82 ! [P: nat > $o,K: nat,B: nat] :
% 0.30/1.82 ( ( P @ K )
% 0.30/1.82 => ( ! [Y2: nat] :
% 0.30/1.82 ( ( P @ Y2 )
% 0.30/1.82 => ( ord_less_eq_nat @ Y2 @ B ) )
% 0.30/1.82 => ? [X2: nat] :
% 0.30/1.82 ( ( P @ X2 )
% 0.30/1.82 & ! [Y3: nat] :
% 0.30/1.82 ( ( P @ Y3 )
% 0.30/1.82 => ( ord_less_eq_nat @ Y3 @ X2 ) ) ) ) ) ).
% 0.30/1.82
% 0.30/1.82 thf(fact_32_le__neq__implies__less,axiom,
% 0.30/1.82 ! [M: nat,N: nat] :
% 0.30/1.82 ( ( ord_less_eq_nat @ M @ N )
% 0.30/1.82 => ( ( M != N )
% 0.30/1.82 => ( ord_less_nat @ M @ N ) ) ) ).
% 0.30/1.82
% 0.30/1.82 thf(fact_33_less__mono__imp__le__mono,axiom,
% 0.30/1.82 ! [F: nat > nat,I: nat,J: nat] :
% 0.30/1.82 ( ! [I2: nat,J2: nat] :
% 0.30/1.82 ( ( ord_less_nat @ I2 @ J2 )
% 0.30/1.82 => ( ord_less_nat @ ( F @ I2 ) @ ( F @ J2 ) ) )
% 0.30/1.82 => ( ( ord_less_eq_nat @ I @ J )
% 0.30/1.82 => ( ord_less_eq_nat @ ( F @ I ) @ ( F @ J ) ) ) ) ).
% 0.30/1.82
% 0.30/1.82 thf(fact_34_length__induct,axiom,
% 0.30/1.82 ! [P: list_c1059388851t_unit > $o,Xs: list_c1059388851t_unit] :
% 0.30/1.82 ( ! [Xs2: list_c1059388851t_unit] :
% 0.30/1.82 ( ! [Ys2: list_c1059388851t_unit] :
% 0.30/1.82 ( ( ord_less_nat @ ( size_s1406904903t_unit @ Ys2 ) @ ( size_s1406904903t_unit @ Xs2 ) )
% 0.30/1.82 => ( P @ Ys2 ) )
% 0.30/1.82 => ( P @ Xs2 ) )
% 0.30/1.82 => ( P @ Xs ) ) ).
% 0.30/1.82
% 0.30/1.82 thf(fact_35_less__imp__diff__less,axiom,
% 0.30/1.82 ! [J: nat,K: nat,N: nat] :
% 0.30/1.82 ( ( ord_less_nat @ J @ K )
% 0.30/1.82 => ( ord_less_nat @ ( minus_minus_nat @ J @ N ) @ K ) ) ).
% 0.30/1.82
% 0.30/1.82 thf(fact_36_diff__less__mono2,axiom,
% 0.30/1.82 ! [M: nat,N: nat,L: nat] :
% 0.30/1.82 ( ( ord_less_nat @ M @ N )
% 0.30/1.82 => ( ( ord_less_nat @ M @ L )
% 0.30/1.82 => ( ord_less_nat @ ( minus_minus_nat @ L @ N ) @ ( minus_minus_nat @ L @ M ) ) ) ) ).
% 0.30/1.82
% 0.30/1.82 thf(fact_37_diff__le__mono2,axiom,
% 0.30/1.82 ! [M: nat,N: nat,L: nat] :
% 0.30/1.82 ( ( ord_less_eq_nat @ M @ N )
% 0.30/1.82 => ( ord_less_eq_nat @ ( minus_minus_nat @ L @ N ) @ ( minus_minus_nat @ L @ M ) ) ) ).
% 0.30/1.82
% 0.30/1.82 thf(fact_38_le__diff__iff_H,axiom,
% 0.30/1.82 ! [A: nat,C: nat,B: nat] :
% 0.30/1.82 ( ( ord_less_eq_nat @ A @ C )
% 0.30/1.82 => ( ( ord_less_eq_nat @ B @ C )
% 0.30/1.82 => ( ( ord_less_eq_nat @ ( minus_minus_nat @ C @ A ) @ ( minus_minus_nat @ C @ B ) )
% 0.30/1.82 = ( ord_less_eq_nat @ B @ A ) ) ) ) ).
% 0.30/1.82
% 0.30/1.82 thf(fact_39_diff__le__self,axiom,
% 0.30/1.82 ! [M: nat,N: nat] : ( ord_less_eq_nat @ ( minus_minus_nat @ M @ N ) @ M ) ).
% 0.30/1.82
% 0.30/1.82 thf(fact_40_diff__le__mono,axiom,
% 0.30/1.82 ! [M: nat,N: nat,L: nat] :
% 0.30/1.82 ( ( ord_less_eq_nat @ M @ N )
% 0.30/1.82 => ( ord_less_eq_nat @ ( minus_minus_nat @ M @ L ) @ ( minus_minus_nat @ N @ L ) ) ) ).
% 0.30/1.82
% 0.30/1.82 thf(fact_41_Nat_Odiff__diff__eq,axiom,
% 0.30/1.82 ! [K: nat,M: nat,N: nat] :
% 0.30/1.82 ( ( ord_less_eq_nat @ K @ M )
% 0.30/1.82 => ( ( ord_less_eq_nat @ K @ N )
% 0.30/1.82 => ( ( minus_minus_nat @ ( minus_minus_nat @ M @ K ) @ ( minus_minus_nat @ N @ K ) )
% 0.30/1.82 = ( minus_minus_nat @ M @ N ) ) ) ) ).
% 0.30/1.82
% 0.30/1.82 thf(fact_42_le__diff__iff,axiom,
% 0.30/1.82 ! [K: nat,M: nat,N: nat] :
% 0.30/1.82 ( ( ord_less_eq_nat @ K @ M )
% 0.30/1.82 => ( ( ord_less_eq_nat @ K @ N )
% 0.30/1.82 => ( ( ord_less_eq_nat @ ( minus_minus_nat @ M @ K ) @ ( minus_minus_nat @ N @ K ) )
% 0.30/1.82 = ( ord_less_eq_nat @ M @ N ) ) ) ) ).
% 0.30/1.82
% 0.30/1.82 thf(fact_43_eq__diff__iff,axiom,
% 0.30/1.82 ! [K: nat,M: nat,N: nat] :
% 0.30/1.82 ( ( ord_less_eq_nat @ K @ M )
% 0.30/1.82 => ( ( ord_less_eq_nat @ K @ N )
% 0.30/1.82 => ( ( ( minus_minus_nat @ M @ K )
% 0.30/1.82 = ( minus_minus_nat @ N @ K ) )
% 0.30/1.82 = ( M = N ) ) ) ) ).
% 0.30/1.82
% 0.30/1.82 thf(fact_44_list__eq__iff__nth__eq,axiom,
% 0.30/1.82 ( ( ^ [Y4: list_c1059388851t_unit,Z: list_c1059388851t_unit] : ( Y4 = Z ) )
% 0.30/1.82 = ( ^ [Xs3: list_c1059388851t_unit,Ys3: list_c1059388851t_unit] :
% 0.30/1.82 ( ( ( size_s1406904903t_unit @ Xs3 )
% 0.30/1.82 = ( size_s1406904903t_unit @ Ys3 ) )
% 0.30/1.82 & ! [I3: nat] :
% 0.30/1.82 ( ( ord_less_nat @ I3 @ ( size_s1406904903t_unit @ Xs3 ) )
% 0.30/1.82 => ( ( nth_co1649820636t_unit @ Xs3 @ I3 )
% 0.30/1.82 = ( nth_co1649820636t_unit @ Ys3 @ I3 ) ) ) ) ) ) ).
% 0.30/1.82
% 0.30/1.82 thf(fact_45_Skolem__list__nth,axiom,
% 0.30/1.82 ! [K: nat,P: nat > config256849571t_unit > $o] :
% 0.30/1.82 ( ( ! [I3: nat] :
% 0.30/1.82 ( ( ord_less_nat @ I3 @ K )
% 0.30/1.82 => ? [X3: config256849571t_unit] : ( P @ I3 @ X3 ) ) )
% 0.30/1.82 = ( ? [Xs3: list_c1059388851t_unit] :
% 0.30/1.82 ( ( ( size_s1406904903t_unit @ Xs3 )
% 0.30/1.82 = K )
% 0.30/1.82 & ! [I3: nat] :
% 0.30/1.82 ( ( ord_less_nat @ I3 @ K )
% 0.30/1.82 => ( P @ I3 @ ( nth_co1649820636t_unit @ Xs3 @ I3 ) ) ) ) ) ) ).
% 0.30/1.82
% 0.30/1.82 thf(fact_46_nth__equalityI,axiom,
% 0.30/1.82 ! [Xs: list_c1059388851t_unit,Ys: list_c1059388851t_unit] :
% 0.30/1.82 ( ( ( size_s1406904903t_unit @ Xs )
% 0.30/1.82 = ( size_s1406904903t_unit @ Ys ) )
% 0.30/1.82 => ( ! [I2: nat] :
% 0.30/1.82 ( ( ord_less_nat @ I2 @ ( size_s1406904903t_unit @ Xs ) )
% 0.30/1.82 => ( ( nth_co1649820636t_unit @ Xs @ I2 )
% 0.30/1.82 = ( nth_co1649820636t_unit @ Ys @ I2 ) ) )
% 0.30/1.82 => ( Xs = Ys ) ) ) ).
% 0.30/1.82
% 0.30/1.82 thf(fact_47_one__reorient,axiom,
% 0.30/1.82 ! [X: nat] :
% 0.30/1.82 ( ( one_one_nat = X )
% 0.30/1.82 = ( X = one_one_nat ) ) ).
% 0.30/1.82
% 0.30/1.82 thf(fact_48_diff__right__commute,axiom,
% 0.30/1.82 ! [A: nat,C: nat,B: nat] :
% 0.30/1.82 ( ( minus_minus_nat @ ( minus_minus_nat @ A @ C ) @ B )
% 0.30/1.82 = ( minus_minus_nat @ ( minus_minus_nat @ A @ B ) @ C ) ) ).
% 0.30/1.82
% 0.30/1.82 thf(fact_49_Length,axiom,
% 0.30/1.82 ! [N4: nat] : ( ord_less_eq_nat @ ( plus_plus_nat @ N4 @ one_one_nat ) @ ( size_s1406904903t_unit @ ( fe @ N4 ) ) ) ).
% 0.30/1.82
% 0.30/1.82 thf(fact_50_AllArePrefixesExec,axiom,
% 0.30/1.82 ! [M3: nat,N4: nat] :
% 0.30/1.82 ( ( ord_less_nat @ M3 @ N4 )
% 0.30/1.82 => ( prefix1615116500t_unit @ ( fe @ M3 ) @ ( fe @ N4 ) ) ) ).
% 0.30/1.82
% 0.30/1.82 thf(fact_51_order__refl,axiom,
% 0.30/1.82 ! [X: nat] : ( ord_less_eq_nat @ X @ X ) ).
% 0.30/1.82
% 0.30/1.82 thf(fact_52_nat__descend__induct,axiom,
% 0.30/1.82 ! [N: nat,P: nat > $o,M: nat] :
% 0.30/1.82 ( ! [K2: nat] :
% 0.30/1.82 ( ( ord_less_nat @ N @ K2 )
% 0.30/1.82 => ( P @ K2 ) )
% 0.30/1.82 => ( ! [K2: nat] :
% 0.30/1.82 ( ( ord_less_eq_nat @ K2 @ N )
% 0.30/1.82 => ( ! [I4: nat] :
% 0.30/1.82 ( ( ord_less_nat @ K2 @ I4 )
% 0.30/1.82 => ( P @ I4 ) )
% 0.30/1.82 => ( P @ K2 ) ) )
% 0.30/1.82 => ( P @ M ) ) ) ).
% 0.30/1.82
% 0.30/1.82 thf(fact_53_less__numeral__extra_I4_J,axiom,
% 0.30/1.82 ~ ( ord_less_nat @ one_one_nat @ one_one_nat ) ).
% 0.30/1.82
% 0.30/1.82 thf(fact_54_le__numeral__extra_I4_J,axiom,
% 0.30/1.82 ord_less_eq_nat @ one_one_nat @ one_one_nat ).
% 0.30/1.82
% 0.30/1.82 thf(fact_55_minf_I8_J,axiom,
% 0.30/1.82 ! [T: nat] :
% 0.30/1.82 ? [Z2: nat] :
% 0.30/1.82 ! [X4: nat] :
% 0.30/1.82 ( ( ord_less_nat @ X4 @ Z2 )
% 0.30/1.82 => ~ ( ord_less_eq_nat @ T @ X4 ) ) ).
% 0.30/1.82
% 0.30/1.82 thf(fact_56_minf_I6_J,axiom,
% 0.30/1.82 ! [T: nat] :
% 0.30/1.82 ? [Z2: nat] :
% 0.30/1.82 ! [X4: nat] :
% 0.30/1.82 ( ( ord_less_nat @ X4 @ Z2 )
% 0.30/1.82 => ( ord_less_eq_nat @ X4 @ T ) ) ).
% 0.30/1.82
% 0.30/1.82 thf(fact_57_pinf_I8_J,axiom,
% 0.30/1.82 ! [T: nat] :
% 0.30/1.82 ? [Z2: nat] :
% 0.30/1.82 ! [X4: nat] :
% 0.30/1.82 ( ( ord_less_nat @ Z2 @ X4 )
% 0.30/1.82 => ( ord_less_eq_nat @ T @ X4 ) ) ).
% 0.30/1.82
% 0.30/1.82 thf(fact_58_pinf_I6_J,axiom,
% 0.30/1.82 ! [T: nat] :
% 0.30/1.82 ? [Z2: nat] :
% 0.30/1.82 ! [X4: nat] :
% 0.30/1.82 ( ( ord_less_nat @ Z2 @ X4 )
% 0.30/1.82 => ~ ( ord_less_eq_nat @ X4 @ T ) ) ).
% 0.30/1.82
% 0.30/1.82 thf(fact_59_add__left__cancel,axiom,
% 0.30/1.82 ! [A: nat,B: nat,C: nat] :
% 0.30/1.82 ( ( ( plus_plus_nat @ A @ B )
% 0.30/1.82 = ( plus_plus_nat @ A @ C ) )
% 0.30/1.82 = ( B = C ) ) ).
% 0.30/1.82
% 0.30/1.82 thf(fact_60_add__right__cancel,axiom,
% 0.30/1.82 ! [B: nat,A: nat,C: nat] :
% 0.30/1.82 ( ( ( plus_plus_nat @ B @ A )
% 0.30/1.82 = ( plus_plus_nat @ C @ A ) )
% 0.30/1.82 = ( B = C ) ) ).
% 0.30/1.82
% 0.30/1.82 thf(fact_61_add__le__cancel__right,axiom,
% 0.30/1.82 ! [A: nat,C: nat,B: nat] :
% 0.30/1.82 ( ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) )
% 0.30/1.82 = ( ord_less_eq_nat @ A @ B ) ) ).
% 0.30/1.82
% 0.30/1.82 thf(fact_62_add__le__cancel__left,axiom,
% 0.30/1.82 ! [C: nat,A: nat,B: nat] :
% 0.30/1.82 ( ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) )
% 0.30/1.82 = ( ord_less_eq_nat @ A @ B ) ) ).
% 0.30/1.82
% 0.30/1.82 thf(fact_63_add__less__cancel__right,axiom,
% 0.30/1.82 ! [A: nat,C: nat,B: nat] :
% 0.30/1.82 ( ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) )
% 0.30/1.82 = ( ord_less_nat @ A @ B ) ) ).
% 0.30/1.82
% 0.30/1.82 thf(fact_64_add__less__cancel__left,axiom,
% 0.30/1.82 ! [C: nat,A: nat,B: nat] :
% 0.30/1.82 ( ( ord_less_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) )
% 0.30/1.82 = ( ord_less_nat @ A @ B ) ) ).
% 0.30/1.82
% 0.30/1.82 thf(fact_65_add__diff__cancel__left,axiom,
% 0.30/1.82 ! [C: nat,A: nat,B: nat] :
% 0.30/1.82 ( ( minus_minus_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) )
% 0.30/1.82 = ( minus_minus_nat @ A @ B ) ) ).
% 0.30/1.82
% 0.30/1.82 thf(fact_66_add__diff__cancel__left_H,axiom,
% 0.30/1.82 ! [A: nat,B: nat] :
% 0.30/1.82 ( ( minus_minus_nat @ ( plus_plus_nat @ A @ B ) @ A )
% 0.30/1.82 = B ) ).
% 0.30/1.82
% 0.30/1.82 thf(fact_67_add__diff__cancel__right,axiom,
% 0.30/1.82 ! [A: nat,C: nat,B: nat] :
% 0.30/1.82 ( ( minus_minus_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) )
% 0.30/1.82 = ( minus_minus_nat @ A @ B ) ) ).
% 0.30/1.82
% 0.30/1.82 thf(fact_68_add__diff__cancel__right_H,axiom,
% 0.30/1.82 ! [A: nat,B: nat] :
% 0.30/1.82 ( ( minus_minus_nat @ ( plus_plus_nat @ A @ B ) @ B )
% 0.30/1.82 = A ) ).
% 0.30/1.82
% 0.30/1.82 thf(fact_69_nat__add__left__cancel__less,axiom,
% 0.30/1.82 ! [K: nat,M: nat,N: nat] :
% 0.30/1.82 ( ( ord_less_nat @ ( plus_plus_nat @ K @ M ) @ ( plus_plus_nat @ K @ N ) )
% 0.30/1.82 = ( ord_less_nat @ M @ N ) ) ).
% 0.30/1.82
% 0.30/1.82 thf(fact_70_nat__add__left__cancel__le,axiom,
% 0.30/1.82 ! [K: nat,M: nat,N: nat] :
% 0.30/1.82 ( ( ord_less_eq_nat @ ( plus_plus_nat @ K @ M ) @ ( plus_plus_nat @ K @ N ) )
% 0.30/1.82 = ( ord_less_eq_nat @ M @ N ) ) ).
% 0.30/1.82
% 0.30/1.82 thf(fact_71_diff__diff__left,axiom,
% 0.30/1.82 ! [I: nat,J: nat,K: nat] :
% 0.30/1.82 ( ( minus_minus_nat @ ( minus_minus_nat @ I @ J ) @ K )
% 0.30/1.82 = ( minus_minus_nat @ I @ ( plus_plus_nat @ J @ K ) ) ) ).
% 0.30/1.82
% 0.30/1.82 thf(fact_72_Nat_Odiff__diff__right,axiom,
% 0.30/1.82 ! [K: nat,J: nat,I: nat] :
% 0.30/1.82 ( ( ord_less_eq_nat @ K @ J )
% 0.30/1.82 => ( ( minus_minus_nat @ I @ ( minus_minus_nat @ J @ K ) )
% 0.30/1.82 = ( minus_minus_nat @ ( plus_plus_nat @ I @ K ) @ J ) ) ) ).
% 0.30/1.82
% 0.30/1.82 thf(fact_73_Nat_Oadd__diff__assoc2,axiom,
% 0.30/1.82 ! [K: nat,J: nat,I: nat] :
% 0.30/1.82 ( ( ord_less_eq_nat @ K @ J )
% 0.30/1.82 => ( ( plus_plus_nat @ ( minus_minus_nat @ J @ K ) @ I )
% 0.30/1.82 = ( minus_minus_nat @ ( plus_plus_nat @ J @ I ) @ K ) ) ) ).
% 0.30/1.82
% 0.30/1.82 thf(fact_74_Nat_Oadd__diff__assoc,axiom,
% 0.30/1.82 ! [K: nat,J: nat,I: nat] :
% 0.30/1.82 ( ( ord_less_eq_nat @ K @ J )
% 0.30/1.82 => ( ( plus_plus_nat @ I @ ( minus_minus_nat @ J @ K ) )
% 0.30/1.82 = ( minus_minus_nat @ ( plus_plus_nat @ I @ J ) @ K ) ) ) ).
% 0.30/1.82
% 0.30/1.82 thf(fact_75_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
% 0.30/1.82 ! [A: nat,B: nat,C: nat] :
% 0.30/1.82 ( ( plus_plus_nat @ ( plus_plus_nat @ A @ B ) @ C )
% 0.30/1.82 = ( plus_plus_nat @ A @ ( plus_plus_nat @ B @ C ) ) ) ).
% 0.30/1.82
% 0.30/1.82 thf(fact_76_add__mono__thms__linordered__semiring_I4_J,axiom,
% 0.30/1.82 ! [I: nat,J: nat,K: nat,L: nat] :
% 0.30/1.82 ( ( ( I = J )
% 0.30/1.82 & ( K = L ) )
% 0.30/1.82 => ( ( plus_plus_nat @ I @ K )
% 0.30/1.82 = ( plus_plus_nat @ J @ L ) ) ) ).
% 0.30/1.82
% 0.30/1.82 thf(fact_77_group__cancel_Oadd1,axiom,
% 0.30/1.82 ! [A2: nat,K: nat,A: nat,B: nat] :
% 0.30/1.82 ( ( A2
% 0.30/1.82 = ( plus_plus_nat @ K @ A ) )
% 0.30/1.82 => ( ( plus_plus_nat @ A2 @ B )
% 0.30/1.82 = ( plus_plus_nat @ K @ ( plus_plus_nat @ A @ B ) ) ) ) ).
% 0.30/1.82
% 0.30/1.82 thf(fact_78_group__cancel_Oadd2,axiom,
% 0.30/1.82 ! [B2: nat,K: nat,B: nat,A: nat] :
% 0.30/1.82 ( ( B2
% 0.30/1.82 = ( plus_plus_nat @ K @ B ) )
% 0.30/1.82 => ( ( plus_plus_nat @ A @ B2 )
% 0.30/1.82 = ( plus_plus_nat @ K @ ( plus_plus_nat @ A @ B ) ) ) ) ).
% 0.30/1.82
% 0.30/1.82 thf(fact_79_add_Oassoc,axiom,
% 0.30/1.82 ! [A: nat,B: nat,C: nat] :
% 0.30/1.82 ( ( plus_plus_nat @ ( plus_plus_nat @ A @ B ) @ C )
% 0.30/1.82 = ( plus_plus_nat @ A @ ( plus_plus_nat @ B @ C ) ) ) ).
% 0.30/1.82
% 0.30/1.82 thf(fact_80_add_Ocommute,axiom,
% 0.30/1.82 ( plus_plus_nat
% 0.30/1.82 = ( ^ [A3: nat,B3: nat] : ( plus_plus_nat @ B3 @ A3 ) ) ) ).
% 0.30/1.82
% 0.30/1.82 thf(fact_81_add_Oleft__commute,axiom,
% 0.30/1.82 ! [B: nat,A: nat,C: nat] :
% 0.30/1.82 ( ( plus_plus_nat @ B @ ( plus_plus_nat @ A @ C ) )
% 0.30/1.82 = ( plus_plus_nat @ A @ ( plus_plus_nat @ B @ C ) ) ) ).
% 0.30/1.82
% 0.30/1.82 thf(fact_82_add__left__imp__eq,axiom,
% 0.30/1.82 ! [A: nat,B: nat,C: nat] :
% 0.30/1.82 ( ( ( plus_plus_nat @ A @ B )
% 0.30/1.82 = ( plus_plus_nat @ A @ C ) )
% 0.30/1.82 => ( B = C ) ) ).
% 0.30/1.82
% 0.30/1.82 thf(fact_83_add__right__imp__eq,axiom,
% 0.30/1.82 ! [B: nat,A: nat,C: nat] :
% 0.30/1.82 ( ( ( plus_plus_nat @ B @ A )
% 0.30/1.82 = ( plus_plus_nat @ C @ A ) )
% 0.30/1.82 => ( B = C ) ) ).
% 0.30/1.82
% 0.30/1.82 thf(fact_84_add__le__imp__le__right,axiom,
% 0.30/1.82 ! [A: nat,C: nat,B: nat] :
% 0.30/1.82 ( ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) )
% 0.30/1.82 => ( ord_less_eq_nat @ A @ B ) ) ).
% 0.30/1.82
% 0.30/1.82 thf(fact_85_add__le__imp__le__left,axiom,
% 0.30/1.82 ! [C: nat,A: nat,B: nat] :
% 0.30/1.82 ( ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) )
% 0.30/1.82 => ( ord_less_eq_nat @ A @ B ) ) ).
% 0.30/1.82
% 0.30/1.82 thf(fact_86_le__iff__add,axiom,
% 0.30/1.82 ( ord_less_eq_nat
% 0.30/1.82 = ( ^ [A3: nat,B3: nat] :
% 0.30/1.82 ? [C2: nat] :
% 0.30/1.82 ( B3
% 0.30/1.82 = ( plus_plus_nat @ A3 @ C2 ) ) ) ) ).
% 0.30/1.82
% 0.30/1.82 thf(fact_87_add__right__mono,axiom,
% 0.30/1.82 ! [A: nat,B: nat,C: nat] :
% 0.30/1.82 ( ( ord_less_eq_nat @ A @ B )
% 0.30/1.82 => ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) ) ) ).
% 0.30/1.82
% 0.30/1.82 thf(fact_88_less__eqE,axiom,
% 0.30/1.82 ! [A: nat,B: nat] :
% 0.30/1.82 ( ( ord_less_eq_nat @ A @ B )
% 0.30/1.82 => ~ ! [C3: nat] :
% 0.30/1.82 ( B
% 0.30/1.82 != ( plus_plus_nat @ A @ C3 ) ) ) ).
% 0.30/1.82
% 0.30/1.82 thf(fact_89_add__left__mono,axiom,
% 0.30/1.82 ! [A: nat,B: nat,C: nat] :
% 0.30/1.82 ( ( ord_less_eq_nat @ A @ B )
% 0.30/1.82 => ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) ) ) ).
% 0.30/1.82
% 0.30/1.82 thf(fact_90_add__mono,axiom,
% 0.30/1.82 ! [A: nat,B: nat,C: nat,D: nat] :
% 0.30/1.82 ( ( ord_less_eq_nat @ A @ B )
% 0.30/1.82 => ( ( ord_less_eq_nat @ C @ D )
% 0.30/1.82 => ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ D ) ) ) ) ).
% 0.30/1.82
% 0.30/1.82 thf(fact_91_add__mono__thms__linordered__semiring_I1_J,axiom,
% 0.30/1.82 ! [I: nat,J: nat,K: nat,L: nat] :
% 0.30/1.82 ( ( ( ord_less_eq_nat @ I @ J )
% 0.30/1.82 & ( ord_less_eq_nat @ K @ L ) )
% 0.30/1.82 => ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% 0.30/1.82
% 0.30/1.82 thf(fact_92_add__mono__thms__linordered__semiring_I2_J,axiom,
% 0.30/1.82 ! [I: nat,J: nat,K: nat,L: nat] :
% 0.30/1.82 ( ( ( I = J )
% 0.30/1.82 & ( ord_less_eq_nat @ K @ L ) )
% 0.30/1.82 => ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% 0.30/1.82
% 0.30/1.82 thf(fact_93_add__mono__thms__linordered__semiring_I3_J,axiom,
% 0.30/1.82 ! [I: nat,J: nat,K: nat,L: nat] :
% 0.30/1.82 ( ( ( ord_less_eq_nat @ I @ J )
% 0.30/1.82 & ( K = L ) )
% 0.30/1.82 => ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% 0.30/1.82
% 0.30/1.82 thf(fact_94_add__less__imp__less__right,axiom,
% 0.30/1.82 ! [A: nat,C: nat,B: nat] :
% 0.30/1.82 ( ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) )
% 0.30/1.82 => ( ord_less_nat @ A @ B ) ) ).
% 0.30/1.82
% 0.30/1.82 thf(fact_95_add__less__imp__less__left,axiom,
% 0.30/1.82 ! [C: nat,A: nat,B: nat] :
% 0.30/1.82 ( ( ord_less_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) )
% 0.30/1.82 => ( ord_less_nat @ A @ B ) ) ).
% 0.30/1.82
% 0.30/1.82 thf(fact_96_add__strict__right__mono,axiom,
% 0.30/1.82 ! [A: nat,B: nat,C: nat] :
% 0.30/1.82 ( ( ord_less_nat @ A @ B )
% 0.30/1.82 => ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) ) ) ).
% 0.30/1.82
% 0.30/1.82 thf(fact_97_add__strict__left__mono,axiom,
% 0.30/1.82 ! [A: nat,B: nat,C: nat] :
% 0.30/1.82 ( ( ord_less_nat @ A @ B )
% 0.30/1.82 => ( ord_less_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) ) ) ).
% 0.30/1.82
% 0.30/1.82 thf(fact_98_add__strict__mono,axiom,
% 0.30/1.82 ! [A: nat,B: nat,C: nat,D: nat] :
% 0.30/1.82 ( ( ord_less_nat @ A @ B )
% 0.30/1.82 => ( ( ord_less_nat @ C @ D )
% 0.30/1.82 => ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ D ) ) ) ) ).
% 0.30/1.82
% 0.30/1.82 thf(fact_99_add__mono__thms__linordered__field_I1_J,axiom,
% 0.30/1.82 ! [I: nat,J: nat,K: nat,L: nat] :
% 0.30/1.82 ( ( ( ord_less_nat @ I @ J )
% 0.30/1.82 & ( K = L ) )
% 0.30/1.82 => ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% 0.30/1.82
% 0.30/1.82 thf(fact_100_add__mono__thms__linordered__field_I2_J,axiom,
% 0.30/1.82 ! [I: nat,J: nat,K: nat,L: nat] :
% 0.30/1.82 ( ( ( I = J )
% 0.30/1.82 & ( ord_less_nat @ K @ L ) )
% 0.30/1.82 => ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% 0.30/1.82
% 0.30/1.82 thf(fact_101_add__mono__thms__linordered__field_I5_J,axiom,
% 0.30/1.82 ! [I: nat,J: nat,K: nat,L: nat] :
% 0.30/1.82 ( ( ( ord_less_nat @ I @ J )
% 0.30/1.82 & ( ord_less_nat @ K @ L ) )
% 0.30/1.82 => ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% 0.30/1.82
% 0.30/1.82 thf(fact_102_diff__diff__add,axiom,
% 0.30/1.82 ! [A: nat,B: nat,C: nat] :
% 0.30/1.82 ( ( minus_minus_nat @ ( minus_minus_nat @ A @ B ) @ C )
% 0.30/1.82 = ( minus_minus_nat @ A @ ( plus_plus_nat @ B @ C ) ) ) ).
% 0.30/1.82
% 0.30/1.82 thf(fact_103_add__implies__diff,axiom,
% 0.30/1.82 ! [C: nat,B: nat,A: nat] :
% 0.30/1.82 ( ( ( plus_plus_nat @ C @ B )
% 0.30/1.82 = A )
% 0.30/1.82 => ( C
% 0.30/1.82 = ( minus_minus_nat @ A @ B ) ) ) ).
% 0.30/1.82
% 0.30/1.82 thf(fact_104_less__add__eq__less,axiom,
% 0.30/1.82 ! [K: nat,L: nat,M: nat,N: nat] :
% 0.30/1.82 ( ( ord_less_nat @ K @ L )
% 0.30/1.82 => ( ( ( plus_plus_nat @ M @ L )
% 0.30/1.82 = ( plus_plus_nat @ K @ N ) )
% 0.30/1.82 => ( ord_less_nat @ M @ N ) ) ) ).
% 0.30/1.82
% 0.30/1.82 thf(fact_105_trans__less__add2,axiom,
% 0.30/1.82 ! [I: nat,J: nat,M: nat] :
% 0.30/1.82 ( ( ord_less_nat @ I @ J )
% 0.30/1.82 => ( ord_less_nat @ I @ ( plus_plus_nat @ M @ J ) ) ) ).
% 0.30/1.82
% 0.30/1.82 thf(fact_106_trans__less__add1,axiom,
% 0.30/1.82 ! [I: nat,J: nat,M: nat] :
% 0.30/1.82 ( ( ord_less_nat @ I @ J )
% 0.30/1.82 => ( ord_less_nat @ I @ ( plus_plus_nat @ J @ M ) ) ) ).
% 0.30/1.82
% 0.30/1.82 thf(fact_107_add__less__mono1,axiom,
% 0.30/1.82 ! [I: nat,J: nat,K: nat] :
% 0.30/1.82 ( ( ord_less_nat @ I @ J )
% 0.30/1.82 => ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ K ) ) ) ).
% 0.30/1.82
% 0.30/1.82 thf(fact_108_not__add__less2,axiom,
% 0.30/1.82 ! [J: nat,I: nat] :
% 0.30/1.82 ~ ( ord_less_nat @ ( plus_plus_nat @ J @ I ) @ I ) ).
% 0.30/1.82
% 0.30/1.82 thf(fact_109_not__add__less1,axiom,
% 0.30/1.82 ! [I: nat,J: nat] :
% 0.30/1.82 ~ ( ord_less_nat @ ( plus_plus_nat @ I @ J ) @ I ) ).
% 0.30/1.82
% 0.30/1.82 thf(fact_110_add__less__mono,axiom,
% 0.30/1.82 ! [I: nat,J: nat,K: nat,L: nat] :
% 0.30/1.82 ( ( ord_less_nat @ I @ J )
% 0.30/1.82 => ( ( ord_less_nat @ K @ L )
% 0.30/1.82 => ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ) ).
% 0.30/1.82
% 0.30/1.82 thf(fact_111_add__lessD1,axiom,
% 0.30/1.82 ! [I: nat,J: nat,K: nat] :
% 0.30/1.82 ( ( ord_less_nat @ ( plus_plus_nat @ I @ J ) @ K )
% 0.30/1.82 => ( ord_less_nat @ I @ K ) ) ).
% 0.30/1.82
% 0.30/1.82 thf(fact_112_nat__le__iff__add,axiom,
% 0.30/1.82 ( ord_less_eq_nat
% 0.30/1.82 = ( ^ [M2: nat,N2: nat] :
% 0.30/1.82 ? [K3: nat] :
% 0.30/1.82 ( N2
% 0.30/1.82 = ( plus_plus_nat @ M2 @ K3 ) ) ) ) ).
% 0.30/1.82
% 0.30/1.82 thf(fact_113_trans__le__add2,axiom,
% 0.30/1.82 ! [I: nat,J: nat,M: nat] :
% 0.30/1.82 ( ( ord_less_eq_nat @ I @ J )
% 0.30/1.82 => ( ord_less_eq_nat @ I @ ( plus_plus_nat @ M @ J ) ) ) ).
% 0.30/1.82
% 0.30/1.82 thf(fact_114_trans__le__add1,axiom,
% 0.30/1.82 ! [I: nat,J: nat,M: nat] :
% 0.30/1.82 ( ( ord_less_eq_nat @ I @ J )
% 0.30/1.82 => ( ord_less_eq_nat @ I @ ( plus_plus_nat @ J @ M ) ) ) ).
% 0.30/1.82
% 0.30/1.82 thf(fact_115_add__le__mono1,axiom,
% 0.30/1.82 ! [I: nat,J: nat,K: nat] :
% 0.30/1.82 ( ( ord_less_eq_nat @ I @ J )
% 0.30/1.82 => ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ K ) ) ) ).
% 0.30/1.82
% 0.30/1.82 thf(fact_116_add__le__mono,axiom,
% 0.30/1.82 ! [I: nat,J: nat,K: nat,L: nat] :
% 0.30/1.82 ( ( ord_less_eq_nat @ I @ J )
% 0.30/1.82 => ( ( ord_less_eq_nat @ K @ L )
% 0.30/1.82 => ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ) ).
% 0.30/1.82
% 0.30/1.82 thf(fact_117_le__Suc__ex,axiom,
% 0.30/1.82 ! [K: nat,L: nat] :
% 0.30/1.82 ( ( ord_less_eq_nat @ K @ L )
% 0.30/1.82 => ? [N3: nat] :
% 0.30/1.82 ( L
% 0.30/1.82 = ( plus_plus_nat @ K @ N3 ) ) ) ).
% 0.30/1.82
% 0.30/1.82 thf(fact_118_add__leD2,axiom,
% 0.30/1.82 ! [M: nat,K: nat,N: nat] :
% 0.30/1.82 ( ( ord_less_eq_nat @ ( plus_plus_nat @ M @ K ) @ N )
% 0.30/1.82 => ( ord_less_eq_nat @ K @ N ) ) ).
% 0.30/1.82
% 0.30/1.82 thf(fact_119_add__leD1,axiom,
% 0.30/1.82 ! [M: nat,K: nat,N: nat] :
% 0.30/1.82 ( ( ord_less_eq_nat @ ( plus_plus_nat @ M @ K ) @ N )
% 0.30/1.82 => ( ord_less_eq_nat @ M @ N ) ) ).
% 0.30/1.82
% 0.30/1.82 thf(fact_120_le__add2,axiom,
% 0.30/1.82 ! [N: nat,M: nat] : ( ord_less_eq_nat @ N @ ( plus_plus_nat @ M @ N ) ) ).
% 0.30/1.82
% 0.30/1.82 thf(fact_121_le__add1,axiom,
% 0.30/1.82 ! [N: nat,M: nat] : ( ord_less_eq_nat @ N @ ( plus_plus_nat @ N @ M ) ) ).
% 0.30/1.82
% 0.30/1.82 thf(fact_122_add__leE,axiom,
% 0.30/1.82 ! [M: nat,K: nat,N: nat] :
% 0.30/1.82 ( ( ord_less_eq_nat @ ( plus_plus_nat @ M @ K ) @ N )
% 0.30/1.82 => ~ ( ( ord_less_eq_nat @ M @ N )
% 0.30/1.82 => ~ ( ord_less_eq_nat @ K @ N ) ) ) ).
% 0.30/1.82
% 0.30/1.82 thf(fact_123_Nat_Odiff__cancel,axiom,
% 0.30/1.82 ! [K: nat,M: nat,N: nat] :
% 0.30/1.82 ( ( minus_minus_nat @ ( plus_plus_nat @ K @ M ) @ ( plus_plus_nat @ K @ N ) )
% 0.30/1.82 = ( minus_minus_nat @ M @ N ) ) ).
% 0.30/1.82
% 0.30/1.82 thf(fact_124_diff__cancel2,axiom,
% 0.30/1.82 ! [M: nat,K: nat,N: nat] :
% 0.30/1.82 ( ( minus_minus_nat @ ( plus_plus_nat @ M @ K ) @ ( plus_plus_nat @ N @ K ) )
% 0.30/1.82 = ( minus_minus_nat @ M @ N ) ) ).
% 0.30/1.82
% 0.30/1.82 thf(fact_125_diff__add__inverse,axiom,
% 0.30/1.82 ! [N: nat,M: nat] :
% 0.30/1.82 ( ( minus_minus_nat @ ( plus_plus_nat @ N @ M ) @ N )
% 0.30/1.82 = M ) ).
% 0.30/1.82
% 0.30/1.82 thf(fact_126_diff__add__inverse2,axiom,
% 0.30/1.82 ! [M: nat,N: nat] :
% 0.30/1.82 ( ( minus_minus_nat @ ( plus_plus_nat @ M @ N ) @ N )
% 0.30/1.82 = M ) ).
% 0.30/1.82
% 0.30/1.82 thf(fact_127_add__less__le__mono,axiom,
% 0.30/1.82 ! [A: nat,B: nat,C: nat,D: nat] :
% 0.30/1.82 ( ( ord_less_nat @ A @ B )
% 0.30/1.82 => ( ( ord_less_eq_nat @ C @ D )
% 0.30/1.82 => ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ D ) ) ) ) ).
% 0.30/1.82
% 0.30/1.82 thf(fact_128_add__le__less__mono,axiom,
% 0.30/1.82 ! [A: nat,B: nat,C: nat,D: nat] :
% 0.30/1.82 ( ( ord_less_eq_nat @ A @ B )
% 0.30/1.82 => ( ( ord_less_nat @ C @ D )
% 0.30/1.82 => ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ D ) ) ) ) ).
% 0.30/1.82
% 0.30/1.82 thf(fact_129_add__mono__thms__linordered__field_I3_J,axiom,
% 0.30/1.82 ! [I: nat,J: nat,K: nat,L: nat] :
% 0.30/1.82 ( ( ( ord_less_nat @ I @ J )
% 0.30/1.82 & ( ord_less_eq_nat @ K @ L ) )
% 0.30/1.82 => ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% 0.30/1.82
% 0.30/1.82 thf(fact_130_add__mono__thms__linordered__field_I4_J,axiom,
% 0.30/1.82 ! [I: nat,J: nat,K: nat,L: nat] :
% 0.30/1.82 ( ( ( ord_less_eq_nat @ I @ J )
% 0.30/1.82 & ( ord_less_nat @ K @ L ) )
% 0.30/1.82 => ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% 0.30/1.82
% 0.30/1.82 thf(fact_131_ordered__cancel__comm__monoid__diff__class_Ole__imp__diff__is__add,axiom,
% 0.30/1.82 ! [A: nat,B: nat,C: nat] :
% 0.30/1.82 ( ( ord_less_eq_nat @ A @ B )
% 0.30/1.82 => ( ( ord_less_eq_nat @ A @ B )
% 0.30/1.82 => ( ( ( minus_minus_nat @ B @ A )
% 0.30/1.82 = C )
% 0.30/1.82 = ( B
% 0.30/1.82 = ( plus_plus_nat @ C @ A ) ) ) ) ) ).
% 0.30/1.82
% 0.30/1.82 thf(fact_132_ordered__cancel__comm__monoid__diff__class_Oadd__diff__inverse,axiom,
% 0.30/1.82 ! [A: nat,B: nat] :
% 0.30/1.82 ( ( ord_less_eq_nat @ A @ B )
% 0.30/1.82 => ( ( plus_plus_nat @ A @ ( minus_minus_nat @ B @ A ) )
% 0.30/1.82 = B ) ) ).
% 0.30/1.82
% 0.30/1.82 thf(fact_133_ordered__cancel__comm__monoid__diff__class_Odiff__diff__right,axiom,
% 0.30/1.82 ! [A: nat,B: nat,C: nat] :
% 0.30/1.82 ( ( ord_less_eq_nat @ A @ B )
% 0.30/1.82 => ( ( minus_minus_nat @ C @ ( minus_minus_nat @ B @ A ) )
% 0.30/1.82 = ( minus_minus_nat @ ( plus_plus_nat @ C @ A ) @ B ) ) ) ).
% 0.30/1.82
% 0.30/1.82 thf(fact_134_ordered__cancel__comm__monoid__diff__class_Odiff__add__assoc2,axiom,
% 0.30/1.82 ! [A: nat,B: nat,C: nat] :
% 0.30/1.82 ( ( ord_less_eq_nat @ A @ B )
% 0.30/1.82 => ( ( minus_minus_nat @ ( plus_plus_nat @ B @ C ) @ A )
% 0.30/1.82 = ( plus_plus_nat @ ( minus_minus_nat @ B @ A ) @ C ) ) ) ).
% 0.30/1.82
% 0.30/1.82 thf(fact_135_ordered__cancel__comm__monoid__diff__class_Oadd__diff__assoc2,axiom,
% 0.30/1.82 ! [A: nat,B: nat,C: nat] :
% 0.30/1.82 ( ( ord_less_eq_nat @ A @ B )
% 0.30/1.82 => ( ( plus_plus_nat @ ( minus_minus_nat @ B @ A ) @ C )
% 0.30/1.82 = ( minus_minus_nat @ ( plus_plus_nat @ B @ C ) @ A ) ) ) ).
% 0.30/1.82
% 0.30/1.82 thf(fact_136_ordered__cancel__comm__monoid__diff__class_Odiff__add__assoc,axiom,
% 0.30/1.82 ! [A: nat,B: nat,C: nat] :
% 0.30/1.82 ( ( ord_less_eq_nat @ A @ B )
% 0.30/1.82 => ( ( minus_minus_nat @ ( plus_plus_nat @ C @ B ) @ A )
% 0.30/1.82 = ( plus_plus_nat @ C @ ( minus_minus_nat @ B @ A ) ) ) ) ).
% 0.30/1.82
% 0.30/1.82 thf(fact_137_ordered__cancel__comm__monoid__diff__class_Oadd__diff__assoc,axiom,
% 0.30/1.82 ! [A: nat,B: nat,C: nat] :
% 0.30/1.82 ( ( ord_less_eq_nat @ A @ B )
% 0.30/1.82 => ( ( plus_plus_nat @ C @ ( minus_minus_nat @ B @ A ) )
% 0.30/1.82 = ( minus_minus_nat @ ( plus_plus_nat @ C @ B ) @ A ) ) ) ).
% 0.30/1.82
% 0.30/1.82 thf(fact_138_ordered__cancel__comm__monoid__diff__class_Ole__diff__conv2,axiom,
% 0.30/1.82 ! [A: nat,B: nat,C: nat] :
% 0.30/1.82 ( ( ord_less_eq_nat @ A @ B )
% 0.30/1.82 => ( ( ord_less_eq_nat @ C @ ( minus_minus_nat @ B @ A ) )
% 0.30/1.82 = ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A ) @ B ) ) ) ).
% 0.30/1.82
% 0.30/1.82 thf(fact_139_le__add__diff,axiom,
% 0.30/1.82 ! [A: nat,B: nat,C: nat] :
% 0.30/1.82 ( ( ord_less_eq_nat @ A @ B )
% 0.30/1.82 => ( ord_less_eq_nat @ C @ ( minus_minus_nat @ ( plus_plus_nat @ B @ C ) @ A ) ) ) ).
% 0.30/1.82
% 0.30/1.82 thf(fact_140_diff__add,axiom,
% 0.30/1.82 ! [A: nat,B: nat] :
% 0.30/1.82 ( ( ord_less_eq_nat @ A @ B )
% 0.30/1.82 => ( ( plus_plus_nat @ ( minus_minus_nat @ B @ A ) @ A )
% 0.30/1.82 = B ) ) ).
% 0.30/1.82
% 0.30/1.82 thf(fact_141_mono__nat__linear__lb,axiom,
% 0.30/1.82 ! [F: nat > nat,M: nat,K: nat] :
% 0.30/1.82 ( ! [M4: nat,N3: nat] :
% 0.30/1.82 ( ( ord_less_nat @ M4 @ N3 )
% 0.30/1.82 => ( ord_less_nat @ ( F @ M4 ) @ ( F @ N3 ) ) )
% 0.30/1.82 => ( ord_less_eq_nat @ ( plus_plus_nat @ ( F @ M ) @ K ) @ ( F @ ( plus_plus_nat @ M @ K ) ) ) ) ).
% 0.30/1.82
% 0.30/1.82 thf(fact_142_add__diff__inverse__nat,axiom,
% 0.30/1.82 ! [M: nat,N: nat] :
% 0.30/1.82 ( ~ ( ord_less_nat @ M @ N )
% 0.30/1.82 => ( ( plus_plus_nat @ N @ ( minus_minus_nat @ M @ N ) )
% 0.30/1.82 = M ) ) ).
% 0.30/1.82
% 0.30/1.82 thf(fact_143_less__diff__conv,axiom,
% 0.30/1.82 ! [I: nat,J: nat,K: nat] :
% 0.30/1.82 ( ( ord_less_nat @ I @ ( minus_minus_nat @ J @ K ) )
% 0.30/1.82 = ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ J ) ) ).
% 0.30/1.82
% 0.30/1.82 thf(fact_144_Nat_Ole__imp__diff__is__add,axiom,
% 0.30/1.82 ! [I: nat,J: nat,K: nat] :
% 0.30/1.82 ( ( ord_less_eq_nat @ I @ J )
% 0.30/1.82 => ( ( ( minus_minus_nat @ J @ I )
% 0.30/1.82 = K )
% 0.30/1.82 = ( J
% 0.30/1.82 = ( plus_plus_nat @ K @ I ) ) ) ) ).
% 0.30/1.82
% 0.30/1.82 thf(fact_145_Nat_Odiff__add__assoc2,axiom,
% 0.30/1.82 ! [K: nat,J: nat,I: nat] :
% 0.30/1.82 ( ( ord_less_eq_nat @ K @ J )
% 0.30/1.82 => ( ( minus_minus_nat @ ( plus_plus_nat @ J @ I ) @ K )
% 0.30/1.82 = ( plus_plus_nat @ ( minus_minus_nat @ J @ K ) @ I ) ) ) ).
% 0.30/1.82
% 0.30/1.82 thf(fact_146_Nat_Odiff__add__assoc,axiom,
% 0.30/1.82 ! [K: nat,J: nat,I: nat] :
% 0.30/1.82 ( ( ord_less_eq_nat @ K @ J )
% 0.30/1.82 => ( ( minus_minus_nat @ ( plus_plus_nat @ I @ J ) @ K )
% 0.30/1.82 = ( plus_plus_nat @ I @ ( minus_minus_nat @ J @ K ) ) ) ) ).
% 0.30/1.82
% 0.30/1.82 thf(fact_147_Nat_Ole__diff__conv2,axiom,
% 0.30/1.82 ! [K: nat,J: nat,I: nat] :
% 0.30/1.82 ( ( ord_less_eq_nat @ K @ J )
% 0.30/1.82 => ( ( ord_less_eq_nat @ I @ ( minus_minus_nat @ J @ K ) )
% 0.30/1.82 = ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ J ) ) ) ).
% 0.30/1.82
% 0.30/1.82 thf(fact_148_le__diff__conv,axiom,
% 0.30/1.82 ! [J: nat,K: nat,I: nat] :
% 0.30/1.82 ( ( ord_less_eq_nat @ ( minus_minus_nat @ J @ K ) @ I )
% 0.30/1.82 = ( ord_less_eq_nat @ J @ ( plus_plus_nat @ I @ K ) ) ) ).
% 0.30/1.82
% 0.30/1.82 thf(fact_149_less__diff__conv2,axiom,
% 0.30/1.82 ! [K: nat,J: nat,I: nat] :
% 0.30/1.82 ( ( ord_less_eq_nat @ K @ J )
% 0.30/1.82 => ( ( ord_less_nat @ ( minus_minus_nat @ J @ K ) @ I )
% 0.30/1.82 = ( ord_less_nat @ J @ ( plus_plus_nat @ I @ K ) ) ) ) ).
% 0.30/1.82
% 0.30/1.82 thf(fact_150_dual__order_Oantisym,axiom,
% 0.30/1.82 ! [B: nat,A: nat] :
% 0.30/1.82 ( ( ord_less_eq_nat @ B @ A )
% 0.30/1.82 => ( ( ord_less_eq_nat @ A @ B )
% 0.30/1.82 => ( A = B ) ) ) ).
% 0.30/1.82
% 0.30/1.82 thf(fact_151_dual__order_Oeq__iff,axiom,
% 0.30/1.82 ( ( ^ [Y4: nat,Z: nat] : ( Y4 = Z ) )
% 0.30/1.82 = ( ^ [A3: nat,B3: nat] :
% 0.30/1.82 ( ( ord_less_eq_nat @ B3 @ A3 )
% 0.30/1.82 & ( ord_less_eq_nat @ A3 @ B3 ) ) ) ) ).
% 0.30/1.82
% 0.30/1.82 thf(fact_152_dual__order_Otrans,axiom,
% 0.30/1.82 ! [B: nat,A: nat,C: nat] :
% 0.30/1.82 ( ( ord_less_eq_nat @ B @ A )
% 0.30/1.82 => ( ( ord_less_eq_nat @ C @ B )
% 0.30/1.82 => ( ord_less_eq_nat @ C @ A ) ) ) ).
% 0.30/1.82
% 0.30/1.82 thf(fact_153_linorder__wlog,axiom,
% 0.30/1.82 ! [P: nat > nat > $o,A: nat,B: nat] :
% 0.30/1.82 ( ! [A4: nat,B4: nat] :
% 0.30/1.82 ( ( ord_less_eq_nat @ A4 @ B4 )
% 0.30/1.82 => ( P @ A4 @ B4 ) )
% 0.30/1.82 => ( ! [A4: nat,B4: nat] :
% 0.30/1.82 ( ( P @ B4 @ A4 )
% 0.30/1.82 => ( P @ A4 @ B4 ) )
% 0.30/1.82 => ( P @ A @ B ) ) ) ).
% 0.30/1.82
% 0.30/1.82 thf(fact_154_dual__order_Orefl,axiom,
% 0.30/1.82 ! [A: nat] : ( ord_less_eq_nat @ A @ A ) ).
% 0.30/1.82
% 0.30/1.82 thf(fact_155_order__trans,axiom,
% 0.30/1.82 ! [X: nat,Y: nat,Z3: nat] :
% 0.30/1.82 ( ( ord_less_eq_nat @ X @ Y )
% 0.30/1.82 => ( ( ord_less_eq_nat @ Y @ Z3 )
% 0.30/1.82 => ( ord_less_eq_nat @ X @ Z3 ) ) ) ).
% 0.30/1.82
% 0.30/1.82 thf(fact_156_order__class_Oorder_Oantisym,axiom,
% 0.30/1.82 ! [A: nat,B: nat] :
% 0.30/1.82 ( ( ord_less_eq_nat @ A @ B )
% 0.30/1.82 => ( ( ord_less_eq_nat @ B @ A )
% 0.30/1.82 => ( A = B ) ) ) ).
% 0.30/1.82
% 0.30/1.82 thf(fact_157_ord__le__eq__trans,axiom,
% 0.30/1.82 ! [A: nat,B: nat,C: nat] :
% 0.30/1.82 ( ( ord_less_eq_nat @ A @ B )
% 0.30/1.82 => ( ( B = C )
% 0.30/1.82 => ( ord_less_eq_nat @ A @ C ) ) ) ).
% 0.30/1.82
% 0.30/1.82 thf(fact_158_ord__eq__le__trans,axiom,
% 0.30/1.82 ! [A: nat,B: nat,C: nat] :
% 0.30/1.82 ( ( A = B )
% 0.30/1.82 => ( ( ord_less_eq_nat @ B @ C )
% 0.30/1.82 => ( ord_less_eq_nat @ A @ C ) ) ) ).
% 0.30/1.82
% 0.30/1.82 thf(fact_159_order__class_Oorder_Oeq__iff,axiom,
% 0.30/1.82 ( ( ^ [Y4: nat,Z: nat] : ( Y4 = Z ) )
% 0.30/1.82 = ( ^ [A3: nat,B3: nat] :
% 0.30/1.82 ( ( ord_less_eq_nat @ A3 @ B3 )
% 0.30/1.82 & ( ord_less_eq_nat @ B3 @ A3 ) ) ) ) ).
% 0.30/1.82
% 0.30/1.82 thf(fact_160_antisym__conv,axiom,
% 0.30/1.82 ! [Y: nat,X: nat] :
% 0.30/1.82 ( ( ord_less_eq_nat @ Y @ X )
% 0.30/1.82 => ( ( ord_less_eq_nat @ X @ Y )
% 0.30/1.82 = ( X = Y ) ) ) ).
% 0.30/1.82
% 0.30/1.82 thf(fact_161_le__cases3,axiom,
% 0.30/1.82 ! [X: nat,Y: nat,Z3: nat] :
% 0.30/1.82 ( ( ( ord_less_eq_nat @ X @ Y )
% 0.30/1.82 => ~ ( ord_less_eq_nat @ Y @ Z3 ) )
% 0.30/1.82 => ( ( ( ord_less_eq_nat @ Y @ X )
% 0.30/1.82 => ~ ( ord_less_eq_nat @ X @ Z3 ) )
% 0.30/1.82 => ( ( ( ord_less_eq_nat @ X @ Z3 )
% 0.30/1.82 => ~ ( ord_less_eq_nat @ Z3 @ Y ) )
% 0.30/1.82 => ( ( ( ord_less_eq_nat @ Z3 @ Y )
% 0.30/1.82 => ~ ( ord_less_eq_nat @ Y @ X ) )
% 0.30/1.82 => ( ( ( ord_less_eq_nat @ Y @ Z3 )
% 0.30/1.82 => ~ ( ord_less_eq_nat @ Z3 @ X ) )
% 0.30/1.82 => ~ ( ( ord_less_eq_nat @ Z3 @ X )
% 0.30/1.82 => ~ ( ord_less_eq_nat @ X @ Y ) ) ) ) ) ) ) ).
% 0.30/1.82
% 0.30/1.82 thf(fact_162_order_Otrans,axiom,
% 0.30/1.82 ! [A: nat,B: nat,C: nat] :
% 0.30/1.82 ( ( ord_less_eq_nat @ A @ B )
% 0.30/1.82 => ( ( ord_less_eq_nat @ B @ C )
% 0.30/1.82 => ( ord_less_eq_nat @ A @ C ) ) ) ).
% 0.30/1.82
% 0.30/1.82 thf(fact_163_le__cases,axiom,
% 0.30/1.82 ! [X: nat,Y: nat] :
% 0.30/1.82 ( ~ ( ord_less_eq_nat @ X @ Y )
% 0.30/1.82 => ( ord_less_eq_nat @ Y @ X ) ) ).
% 0.30/1.82
% 0.30/1.82 thf(fact_164_eq__refl,axiom,
% 0.30/1.82 ! [X: nat,Y: nat] :
% 0.30/1.82 ( ( X = Y )
% 0.30/1.82 => ( ord_less_eq_nat @ X @ Y ) ) ).
% 0.30/1.82
% 0.30/1.82 thf(fact_165_linear,axiom,
% 0.30/1.82 ! [X: nat,Y: nat] :
% 0.30/1.82 ( ( ord_less_eq_nat @ X @ Y )
% 0.30/1.82 | ( ord_less_eq_nat @ Y @ X ) ) ).
% 0.30/1.82
% 0.30/1.82 thf(fact_166_antisym,axiom,
% 0.30/1.82 ! [X: nat,Y: nat] :
% 0.30/1.82 ( ( ord_less_eq_nat @ X @ Y )
% 0.30/1.82 => ( ( ord_less_eq_nat @ Y @ X )
% 0.30/1.82 => ( X = Y ) ) ) ).
% 0.30/1.82
% 0.30/1.82 thf(fact_167_eq__iff,axiom,
% 0.30/1.82 ( ( ^ [Y4: nat,Z: nat] : ( Y4 = Z ) )
% 0.30/1.82 = ( ^ [X5: nat,Y5: nat] :
% 0.30/1.82 ( ( ord_less_eq_nat @ X5 @ Y5 )
% 0.30/1.82 & ( ord_less_eq_nat @ Y5 @ X5 ) ) ) ) ).
% 0.30/1.82
% 0.30/1.82 thf(fact_168_ord__le__eq__subst,axiom,
% 0.30/1.82 ! [A: nat,B: nat,F: nat > nat,C: nat] :
% 0.30/1.82 ( ( ord_less_eq_nat @ A @ B )
% 0.30/1.82 => ( ( ( F @ B )
% 0.30/1.82 = C )
% 0.30/1.82 => ( ! [X2: nat,Y2: nat] :
% 0.30/1.82 ( ( ord_less_eq_nat @ X2 @ Y2 )
% 0.30/1.82 => ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
% 0.30/1.82 => ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% 0.30/1.82
% 0.30/1.82 thf(fact_169_ord__eq__le__subst,axiom,
% 0.30/1.82 ! [A: nat,F: nat > nat,B: nat,C: nat] :
% 0.30/1.82 ( ( A
% 0.30/1.82 = ( F @ B ) )
% 0.30/1.82 => ( ( ord_less_eq_nat @ B @ C )
% 0.30/1.82 => ( ! [X2: nat,Y2: nat] :
% 0.30/1.82 ( ( ord_less_eq_nat @ X2 @ Y2 )
% 0.30/1.82 => ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
% 0.30/1.82 => ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% 0.30/1.82
% 0.30/1.82 thf(fact_170_order__subst2,axiom,
% 0.30/1.82 ! [A: nat,B: nat,F: nat > nat,C: nat] :
% 0.30/1.82 ( ( ord_less_eq_nat @ A @ B )
% 0.30/1.82 => ( ( ord_less_eq_nat @ ( F @ B ) @ C )
% 0.30/1.82 => ( ! [X2: nat,Y2: nat] :
% 0.30/1.82 ( ( ord_less_eq_nat @ X2 @ Y2 )
% 0.30/1.82 => ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
% 0.30/1.82 => ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% 0.30/1.82
% 0.30/1.82 thf(fact_171_order__subst1,axiom,
% 0.30/1.82 ! [A: nat,F: nat > nat,B: nat,C: nat] :
% 0.30/1.82 ( ( ord_less_eq_nat @ A @ ( F @ B ) )
% 0.30/1.82 => ( ( ord_less_eq_nat @ B @ C )
% 0.30/1.82 => ( ! [X2: nat,Y2: nat] :
% 0.30/1.82 ( ( ord_less_eq_nat @ X2 @ Y2 )
% 0.30/1.82 => ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
% 0.30/1.82 => ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% 0.30/1.82
% 0.30/1.82 thf(fact_172_dual__order_Ostrict__implies__not__eq,axiom,
% 0.30/1.82 ! [B: nat,A: nat] :
% 0.30/1.82 ( ( ord_less_nat @ B @ A )
% 0.30/1.82 => ( A != B ) ) ).
% 0.30/1.82
% 0.30/1.82 thf(fact_173_order_Ostrict__implies__not__eq,axiom,
% 0.30/1.82 ! [A: nat,B: nat] :
% 0.30/1.82 ( ( ord_less_nat @ A @ B )
% 0.30/1.82 => ( A != B ) ) ).
% 0.30/1.82
% 0.30/1.82 thf(fact_174_not__less__iff__gr__or__eq,axiom,
% 0.30/1.82 ! [X: nat,Y: nat] :
% 0.30/1.82 ( ( ~ ( ord_less_nat @ X @ Y ) )
% 0.30/1.82 = ( ( ord_less_nat @ Y @ X )
% 0.30/1.82 | ( X = Y ) ) ) ).
% 0.30/1.82
% 0.30/1.82 thf(fact_175_dual__order_Ostrict__trans,axiom,
% 0.30/1.82 ! [B: nat,A: nat,C: nat] :
% 0.30/1.82 ( ( ord_less_nat @ B @ A )
% 0.30/1.82 => ( ( ord_less_nat @ C @ B )
% 0.30/1.82 => ( ord_less_nat @ C @ A ) ) ) ).
% 0.30/1.82
% 0.30/1.82 thf(fact_176_linorder__less__wlog,axiom,
% 0.30/1.82 ! [P: nat > nat > $o,A: nat,B: nat] :
% 0.30/1.82 ( ! [A4: nat,B4: nat] :
% 0.30/1.82 ( ( ord_less_nat @ A4 @ B4 )
% 0.30/1.82 => ( P @ A4 @ B4 ) )
% 0.30/1.82 => ( ! [A4: nat] : ( P @ A4 @ A4 )
% 0.30/1.82 => ( ! [A4: nat,B4: nat] :
% 0.30/1.82 ( ( P @ B4 @ A4 )
% 0.30/1.82 => ( P @ A4 @ B4 ) )
% 0.30/1.82 => ( P @ A @ B ) ) ) ) ).
% 0.30/1.82
% 0.30/1.82 thf(fact_177_exists__least__iff,axiom,
% 0.30/1.82 ( ( ^ [P2: nat > $o] :
% 0.30/1.82 ? [X3: nat] : ( P2 @ X3 ) )
% 0.30/1.82 = ( ^ [P3: nat > $o] :
% 0.30/1.82 ? [N2: nat] :
% 0.30/1.82 ( ( P3 @ N2 )
% 0.30/1.82 & ! [M2: nat] :
% 0.30/1.82 ( ( ord_less_nat @ M2 @ N2 )
% 0.30/1.82 => ~ ( P3 @ M2 ) ) ) ) ) ).
% 0.30/1.82
% 0.30/1.82 thf(fact_178_less__imp__not__less,axiom,
% 0.30/1.82 ! [X: nat,Y: nat] :
% 0.30/1.82 ( ( ord_less_nat @ X @ Y )
% 0.30/1.82 => ~ ( ord_less_nat @ Y @ X ) ) ).
% 0.30/1.82
% 0.30/1.82 thf(fact_179_order_Ostrict__trans,axiom,
% 0.30/1.82 ! [A: nat,B: nat,C: nat] :
% 0.30/1.82 ( ( ord_less_nat @ A @ B )
% 0.30/1.82 => ( ( ord_less_nat @ B @ C )
% 0.30/1.82 => ( ord_less_nat @ A @ C ) ) ) ).
% 0.30/1.82
% 0.30/1.82 thf(fact_180_dual__order_Oirrefl,axiom,
% 0.30/1.82 ! [A: nat] :
% 0.30/1.82 ~ ( ord_less_nat @ A @ A ) ).
% 0.30/1.82
% 0.30/1.82 thf(fact_181_linorder__cases,axiom,
% 0.30/1.82 ! [X: nat,Y: nat] :
% 0.30/1.82 ( ~ ( ord_less_nat @ X @ Y )
% 0.30/1.82 => ( ( X != Y )
% 0.30/1.82 => ( ord_less_nat @ Y @ X ) ) ) ).
% 0.30/1.82
% 0.30/1.82 thf(fact_182_less__imp__triv,axiom,
% 0.30/1.82 ! [X: nat,Y: nat,P: $o] :
% 0.30/1.82 ( ( ord_less_nat @ X @ Y )
% 0.30/1.82 => ( ( ord_less_nat @ Y @ X )
% 0.30/1.82 => P ) ) ).
% 0.30/1.82
% 0.30/1.82 thf(fact_183_less__imp__not__eq2,axiom,
% 0.30/1.82 ! [X: nat,Y: nat] :
% 0.30/1.82 ( ( ord_less_nat @ X @ Y )
% 0.30/1.82 => ( Y != X ) ) ).
% 0.30/1.82
% 0.30/1.82 thf(fact_184_antisym__conv3,axiom,
% 0.30/1.82 ! [Y: nat,X: nat] :
% 0.30/1.82 ( ~ ( ord_less_nat @ Y @ X )
% 0.30/1.82 => ( ( ~ ( ord_less_nat @ X @ Y ) )
% 0.30/1.82 = ( X = Y ) ) ) ).
% 0.30/1.82
% 0.30/1.82 thf(fact_185_less__induct,axiom,
% 0.30/1.82 ! [P: nat > $o,A: nat] :
% 0.30/1.82 ( ! [X2: nat] :
% 0.30/1.82 ( ! [Y3: nat] :
% 0.30/1.82 ( ( ord_less_nat @ Y3 @ X2 )
% 0.30/1.82 => ( P @ Y3 ) )
% 0.30/1.82 => ( P @ X2 ) )
% 0.30/1.82 => ( P @ A ) ) ).
% 0.30/1.82
% 0.30/1.82 thf(fact_186_less__not__sym,axiom,
% 0.30/1.82 ! [X: nat,Y: nat] :
% 0.30/1.82 ( ( ord_less_nat @ X @ Y )
% 0.30/1.82 => ~ ( ord_less_nat @ Y @ X ) ) ).
% 0.30/1.82
% 0.30/1.82 thf(fact_187_less__imp__not__eq,axiom,
% 0.30/1.82 ! [X: nat,Y: nat] :
% 0.30/1.82 ( ( ord_less_nat @ X @ Y )
% 0.30/1.82 => ( X != Y ) ) ).
% 0.30/1.82
% 0.30/1.82 thf(fact_188_dual__order_Oasym,axiom,
% 0.30/1.82 ! [B: nat,A: nat] :
% 0.30/1.82 ( ( ord_less_nat @ B @ A )
% 0.30/1.82 => ~ ( ord_less_nat @ A @ B ) ) ).
% 0.30/1.82
% 0.30/1.82 thf(fact_189_ord__less__eq__trans,axiom,
% 0.30/1.82 ! [A: nat,B: nat,C: nat] :
% 0.30/1.82 ( ( ord_less_nat @ A @ B )
% 0.30/1.82 => ( ( B = C )
% 0.30/1.82 => ( ord_less_nat @ A @ C ) ) ) ).
% 0.30/1.82
% 0.30/1.82 thf(fact_190_ord__eq__less__trans,axiom,
% 0.30/1.82 ! [A: nat,B: nat,C: nat] :
% 0.30/1.82 ( ( A = B )
% 0.30/1.82 => ( ( ord_less_nat @ B @ C )
% 0.30/1.82 => ( ord_less_nat @ A @ C ) ) ) ).
% 0.30/1.82
% 0.30/1.82 thf(fact_191_less__irrefl,axiom,
% 0.30/1.82 ! [X: nat] :
% 0.30/1.82 ~ ( ord_less_nat @ X @ X ) ).
% 0.30/1.82
% 0.30/1.82 thf(fact_192_less__linear,axiom,
% 0.30/1.82 ! [X: nat,Y: nat] :
% 0.30/1.82 ( ( ord_less_nat @ X @ Y )
% 0.30/1.82 | ( X = Y )
% 0.30/1.82 | ( ord_less_nat @ Y @ X ) ) ).
% 0.30/1.82
% 0.30/1.82 thf(fact_193_less__trans,axiom,
% 0.30/1.82 ! [X: nat,Y: nat,Z3: nat] :
% 0.30/1.82 ( ( ord_less_nat @ X @ Y )
% 0.30/1.82 => ( ( ord_less_nat @ Y @ Z3 )
% 0.30/1.82 => ( ord_less_nat @ X @ Z3 ) ) ) ).
% 0.30/1.82
% 0.30/1.82 thf(fact_194_less__asym_H,axiom,
% 0.30/1.82 ! [A: nat,B: nat] :
% 0.30/1.82 ( ( ord_less_nat @ A @ B )
% 0.30/1.82 => ~ ( ord_less_nat @ B @ A ) ) ).
% 0.30/1.82
% 0.30/1.82 thf(fact_195_less__asym,axiom,
% 0.30/1.82 ! [X: nat,Y: nat] :
% 0.30/1.82 ( ( ord_less_nat @ X @ Y )
% 0.30/1.82 => ~ ( ord_less_nat @ Y @ X ) ) ).
% 0.30/1.82
% 0.30/1.82 thf(fact_196_less__imp__neq,axiom,
% 0.30/1.82 ! [X: nat,Y: nat] :
% 0.30/1.82 ( ( ord_less_nat @ X @ Y )
% 0.30/1.82 => ( X != Y ) ) ).
% 0.30/1.82
% 0.30/1.82 thf(fact_197_order_Oasym,axiom,
% 0.30/1.82 ! [A: nat,B: nat] :
% 0.30/1.82 ( ( ord_less_nat @ A @ B )
% 0.30/1.82 => ~ ( ord_less_nat @ B @ A ) ) ).
% 0.30/1.82
% 0.30/1.82 thf(fact_198_neq__iff,axiom,
% 0.30/1.82 ! [X: nat,Y: nat] :
% 0.30/1.82 ( ( X != Y )
% 0.30/1.82 = ( ( ord_less_nat @ X @ Y )
% 0.30/1.82 | ( ord_less_nat @ Y @ X ) ) ) ).
% 0.30/1.82
% 0.30/1.82 thf(fact_199_neqE,axiom,
% 0.30/1.82 ! [X: nat,Y: nat] :
% 0.30/1.82 ( ( X != Y )
% 0.30/1.82 => ( ~ ( ord_less_nat @ X @ Y )
% 0.30/1.82 => ( ord_less_nat @ Y @ X ) ) ) ).
% 0.30/1.82
% 0.30/1.82 thf(fact_200_gt__ex,axiom,
% 0.30/1.82 ! [X: nat] :
% 0.30/1.82 ? [X_1: nat] : ( ord_less_nat @ X @ X_1 ) ).
% 0.30/1.82
% 0.30/1.82 thf(fact_201_order__less__subst2,axiom,
% 0.30/1.82 ! [A: nat,B: nat,F: nat > nat,C: nat] :
% 0.30/1.82 ( ( ord_less_nat @ A @ B )
% 0.30/1.82 => ( ( ord_less_nat @ ( F @ B ) @ C )
% 0.30/1.82 => ( ! [X2: nat,Y2: nat] :
% 0.30/1.82 ( ( ord_less_nat @ X2 @ Y2 )
% 0.30/1.82 => ( ord_less_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
% 0.30/1.82 => ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% 0.30/1.82
% 0.30/1.82 thf(fact_202_order__less__subst1,axiom,
% 0.30/1.82 ! [A: nat,F: nat > nat,B: nat,C: nat] :
% 0.30/1.82 ( ( ord_less_nat @ A @ ( F @ B ) )
% 0.30/1.82 => ( ( ord_less_nat @ B @ C )
% 0.30/1.82 => ( ! [X2: nat,Y2: nat] :
% 0.30/1.82 ( ( ord_less_nat @ X2 @ Y2 )
% 0.30/1.82 => ( ord_less_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
% 0.30/1.82 => ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% 0.30/1.82
% 0.30/1.82 thf(fact_203_ord__less__eq__subst,axiom,
% 0.30/1.82 ! [A: nat,B: nat,F: nat > nat,C: nat] :
% 0.30/1.82 ( ( ord_less_nat @ A @ B )
% 0.30/1.82 => ( ( ( F @ B )
% 0.30/1.82 = C )
% 0.30/1.82 => ( ! [X2: nat,Y2: nat] :
% 0.30/1.82 ( ( ord_less_nat @ X2 @ Y2 )
% 0.30/1.82 => ( ord_less_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
% 0.30/1.82 => ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% 0.30/1.82
% 0.30/1.82 thf(fact_204_ord__eq__less__subst,axiom,
% 0.30/1.82 ! [A: nat,F: nat > nat,B: nat,C: nat] :
% 0.30/1.82 ( ( A
% 0.30/1.82 = ( F @ B ) )
% 0.30/1.82 => ( ( ord_less_nat @ B @ C )
% 0.30/1.82 => ( ! [X2: nat,Y2: nat] :
% 0.30/1.82 ( ( ord_less_nat @ X2 @ Y2 )
% 0.30/1.82 => ( ord_less_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
% 0.30/1.82 => ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% 0.30/1.82
% 0.30/1.82 thf(fact_205_pinf_I1_J,axiom,
% 0.30/1.82 ! [P: nat > $o,P4: nat > $o,Q: nat > $o,Q2: nat > $o] :
% 0.30/1.82 ( ? [Z4: nat] :
% 0.30/1.82 ! [X2: nat] :
% 0.30/1.82 ( ( ord_less_nat @ Z4 @ X2 )
% 0.30/1.82 => ( ( P @ X2 )
% 0.30/1.82 = ( P4 @ X2 ) ) )
% 0.30/1.82 => ( ? [Z4: nat] :
% 0.30/1.82 ! [X2: nat] :
% 0.30/1.82 ( ( ord_less_nat @ Z4 @ X2 )
% 0.30/1.82 => ( ( Q @ X2 )
% 0.30/1.82 = ( Q2 @ X2 ) ) )
% 0.30/1.82 => ? [Z2: nat] :
% 0.30/1.82 ! [X4: nat] :
% 0.30/1.82 ( ( ord_less_nat @ Z2 @ X4 )
% 0.30/1.82 => ( ( ( P @ X4 )
% 0.30/1.82 & ( Q @ X4 ) )
% 0.30/1.82 = ( ( P4 @ X4 )
% 0.30/1.82 & ( Q2 @ X4 ) ) ) ) ) ) ).
% 0.30/1.82
% 0.30/1.82 thf(fact_206_pinf_I2_J,axiom,
% 0.30/1.82 ! [P: nat > $o,P4: nat > $o,Q: nat > $o,Q2: nat > $o] :
% 0.30/1.82 ( ? [Z4: nat] :
% 0.30/1.82 ! [X2: nat] :
% 0.30/1.82 ( ( ord_less_nat @ Z4 @ X2 )
% 0.30/1.82 => ( ( P @ X2 )
% 0.30/1.82 = ( P4 @ X2 ) ) )
% 0.30/1.82 => ( ? [Z4: nat] :
% 0.30/1.82 ! [X2: nat] :
% 0.30/1.82 ( ( ord_less_nat @ Z4 @ X2 )
% 0.30/1.82 => ( ( Q @ X2 )
% 0.30/1.82 = ( Q2 @ X2 ) ) )
% 0.30/1.82 => ? [Z2: nat] :
% 0.30/1.82 ! [X4: nat] :
% 0.30/1.82 ( ( ord_less_nat @ Z2 @ X4 )
% 0.30/1.82 => ( ( ( P @ X4 )
% 0.30/1.82 | ( Q @ X4 ) )
% 0.30/1.82 = ( ( P4 @ X4 )
% 0.30/1.82 | ( Q2 @ X4 ) ) ) ) ) ) ).
% 0.30/1.82
% 0.30/1.82 thf(fact_207_pinf_I3_J,axiom,
% 0.30/1.82 ! [T: nat] :
% 0.30/1.82 ? [Z2: nat] :
% 0.30/1.82 ! [X4: nat] :
% 0.30/1.82 ( ( ord_less_nat @ Z2 @ X4 )
% 0.30/1.82 => ( X4 != T ) ) ).
% 0.30/1.82
% 0.30/1.82 thf(fact_208_pinf_I4_J,axiom,
% 0.30/1.82 ! [T: nat] :
% 0.30/1.82 ? [Z2: nat] :
% 0.30/1.82 ! [X4: nat] :
% 0.30/1.82 ( ( ord_less_nat @ Z2 @ X4 )
% 0.30/1.82 => ( X4 != T ) ) ).
% 0.30/1.82
% 0.30/1.82 thf(fact_209_pinf_I5_J,axiom,
% 0.30/1.82 ! [T: nat] :
% 0.30/1.82 ? [Z2: nat] :
% 0.30/1.82 ! [X4: nat] :
% 0.30/1.82 ( ( ord_less_nat @ Z2 @ X4 )
% 0.30/1.82 => ~ ( ord_less_nat @ X4 @ T ) ) ).
% 0.30/1.82
% 0.30/1.82 thf(fact_210_pinf_I7_J,axiom,
% 0.30/1.82 ! [T: nat] :
% 0.30/1.82 ? [Z2: nat] :
% 0.30/1.82 ! [X4: nat] :
% 0.30/1.82 ( ( ord_less_nat @ Z2 @ X4 )
% 0.30/1.82 => ( ord_less_nat @ T @ X4 ) ) ).
% 0.30/1.82
% 0.30/1.82 thf(fact_211_minf_I1_J,axiom,
% 0.30/1.82 ! [P: nat > $o,P4: nat > $o,Q: nat > $o,Q2: nat > $o] :
% 0.30/1.82 ( ? [Z4: nat] :
% 0.30/1.82 ! [X2: nat] :
% 0.30/1.82 ( ( ord_less_nat @ X2 @ Z4 )
% 0.30/1.82 => ( ( P @ X2 )
% 0.30/1.82 = ( P4 @ X2 ) ) )
% 0.30/1.82 => ( ? [Z4: nat] :
% 0.30/1.82 ! [X2: nat] :
% 0.30/1.82 ( ( ord_less_nat @ X2 @ Z4 )
% 0.30/1.82 => ( ( Q @ X2 )
% 0.30/1.82 = ( Q2 @ X2 ) ) )
% 0.30/1.82 => ? [Z2: nat] :
% 0.30/1.82 ! [X4: nat] :
% 0.30/1.82 ( ( ord_less_nat @ X4 @ Z2 )
% 0.30/1.82 => ( ( ( P @ X4 )
% 0.30/1.82 & ( Q @ X4 ) )
% 0.30/1.82 = ( ( P4 @ X4 )
% 0.30/1.82 & ( Q2 @ X4 ) ) ) ) ) ) ).
% 0.30/1.82
% 0.30/1.82 thf(fact_212_minf_I2_J,axiom,
% 0.30/1.82 ! [P: nat > $o,P4: nat > $o,Q: nat > $o,Q2: nat > $o] :
% 0.30/1.82 ( ? [Z4: nat] :
% 0.30/1.82 ! [X2: nat] :
% 0.30/1.82 ( ( ord_less_nat @ X2 @ Z4 )
% 0.30/1.82 => ( ( P @ X2 )
% 0.30/1.82 = ( P4 @ X2 ) ) )
% 0.30/1.82 => ( ? [Z4: nat] :
% 0.30/1.82 ! [X2: nat] :
% 0.30/1.82 ( ( ord_less_nat @ X2 @ Z4 )
% 0.30/1.82 => ( ( Q @ X2 )
% 0.30/1.82 = ( Q2 @ X2 ) ) )
% 0.30/1.82 => ? [Z2: nat] :
% 0.30/1.82 ! [X4: nat] :
% 0.30/1.82 ( ( ord_less_nat @ X4 @ Z2 )
% 0.30/1.82 => ( ( ( P @ X4 )
% 0.30/1.82 | ( Q @ X4 ) )
% 0.30/1.82 = ( ( P4 @ X4 )
% 0.30/1.82 | ( Q2 @ X4 ) ) ) ) ) ) ).
% 0.30/1.82
% 0.30/1.82 thf(fact_213_minf_I3_J,axiom,
% 0.30/1.82 ! [T: nat] :
% 0.30/1.82 ? [Z2: nat] :
% 0.30/1.82 ! [X4: nat] :
% 0.30/1.82 ( ( ord_less_nat @ X4 @ Z2 )
% 0.30/1.82 => ( X4 != T ) ) ).
% 0.30/1.82
% 0.30/1.82 thf(conj_0,conjecture,
% 0.30/1.82 ( ( last_c571238084t_unit @ ( fe @ n ) )
% 0.30/1.82 = ( nth_co1649820636t_unit @ ( fe @ n ) @ ( minus_minus_nat @ ( size_s1406904903t_unit @ ( fe @ n ) ) @ one_one_nat ) ) ) ).
% 0.30/1.82 ------------------------
% 0.30/1.84 % (32322)lrs+10_1:1_au=on:inj=on:i=2:si=on:rtra=on_0 on DTF2THF_32211 for (2999ds/2Mi)
% 0.30/1.84 % (32318)lrs+1002_1:8_bd=off:fd=off:hud=10:tnu=1:i=183:si=on:rtra=on_0 on DTF2THF_32211 for (2999ds/183Mi)
% 0.30/1.84 % (32321)dis+1010_1:1_au=on:cbe=off:chr=on:fsr=off:hfsq=on:nm=64:sos=theory:sp=weighted_frequency:i=27:si=on:rtra=on_0 on DTF2THF_32211 for (2999ds/27Mi)
% 0.30/1.84 % (32319)lrs+10_1:1_c=on:cnfonf=conj_eager:fd=off:fe=off:kws=frequency:spb=intro:i=4:si=on:rtra=on_0 on DTF2THF_32211 for (2999ds/4Mi)
% 0.30/1.84 % (32325)lrs+1004_1:128_cond=on:e2e=on:sp=weighted_frequency:i=18:si=on:rtra=on_0 on DTF2THF_32211 for (2999ds/18Mi)
% 0.30/1.84 % (32323)lrs+1002_1:128_aac=none:au=on:cnfonf=lazy_not_gen_be_off:sos=all:i=2:si=on:rtra=on_0 on DTF2THF_32211 for (2999ds/2Mi)
% 0.30/1.84 % (32324)lrs+1002_1:1_au=on:bd=off:e2e=on:sd=2:sos=on:ss=axioms:i=275:si=on:rtra=on_0 on DTF2THF_32211 for (2999ds/275Mi)
% 0.30/1.84 % (32322)Instruction limit reached!
% 0.30/1.84 % (32322)------------------------------
% 0.30/1.84 % (32322)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 0.30/1.84 % (32322)Termination reason: Unknown
% 0.30/1.84 % (32322)Termination phase: shuffling
% 0.30/1.84
% 0.30/1.84 % (32322)Memory used [KB]: 1279
% 0.30/1.84 % (32322)Time elapsed: 0.002 s
% 0.30/1.84 % (32322)Instructions burned: 2 (million)
% 0.30/1.84 % (32322)------------------------------
% 0.30/1.84 % (32322)------------------------------
% 0.30/1.84 % (32323)Instruction limit reached!
% 0.30/1.84 % (32323)------------------------------
% 0.30/1.84 % (32323)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 0.30/1.84 % (32323)Termination reason: Unknown
% 0.30/1.84 % (32323)Termination phase: shuffling
% 0.30/1.84
% 0.30/1.84 % (32323)Memory used [KB]: 1279
% 0.30/1.84 % (32323)Time elapsed: 0.002 s
% 0.30/1.84 % (32323)Instructions burned: 2 (million)
% 0.30/1.84 % (32323)------------------------------
% 0.30/1.84 % (32323)------------------------------
% 0.30/1.84 % (32319)Instruction limit reached!
% 0.30/1.84 % (32319)------------------------------
% 0.30/1.84 % (32319)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 0.30/1.84 % (32319)Termination reason: Unknown
% 0.30/1.84 % (32319)Termination phase: shuffling
% 0.30/1.84
% 0.30/1.84 % (32319)Memory used [KB]: 1279
% 0.30/1.84 % (32319)Time elapsed: 0.003 s
% 0.30/1.84 % (32319)Instructions burned: 5 (million)
% 0.30/1.84 % (32319)------------------------------
% 0.30/1.84 % (32319)------------------------------
% 0.30/1.84 % (32325)Instruction limit reached!
% 0.30/1.84 % (32325)------------------------------
% 0.30/1.84 % (32325)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 0.30/1.84 % (32325)Termination reason: Unknown
% 0.30/1.84 % (32325)Termination phase: Property scanning
% 0.30/1.84
% 0.30/1.84 % (32325)Memory used [KB]: 1535
% 0.30/1.84 % (32325)Time elapsed: 0.007 s
% 0.30/1.84 % (32325)Instructions burned: 18 (million)
% 0.30/1.84 % (32325)------------------------------
% 0.30/1.84 % (32325)------------------------------
% 0.30/1.84 % (32321)Instruction limit reached!
% 0.30/1.84 % (32321)------------------------------
% 0.30/1.84 % (32321)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 0.30/1.84 % (32321)Termination reason: Unknown
% 0.30/1.84 % (32321)Termination phase: Preprocessing 3
% 0.30/1.84
% 0.30/1.84 % (32321)Memory used [KB]: 1663
% 0.30/1.84 % (32321)Time elapsed: 0.010 s
% 0.30/1.84 % (32321)Instructions burned: 28 (million)
% 0.30/1.84 % (32321)------------------------------
% 0.30/1.84 % (32321)------------------------------
% 0.30/1.85 % (32329)lrs+10_1:1_bet=on:cnfonf=off:fd=off:hud=5:inj=on:i=3:si=on:rtra=on_0 on DTF2THF_32211 for (2999ds/3Mi)
% 0.30/1.85 % (32330)lrs+1002_1:1_cnfonf=lazy_not_be_gen:hud=14:prag=on:sp=weighted_frequency:tnu=1:i=37:si=on:rtra=on_0 on DTF2THF_32211 for (2999ds/37Mi)
% 0.30/1.85 % (32332)lrs+2_16:1_acc=model:au=on:bd=off:c=on:e2e=on:nm=2:sos=all:i=15:si=on:rtra=on_0 on DTF2THF_32211 for (2999ds/15Mi)
% 0.30/1.85 % (32329)Instruction limit reached!
% 0.30/1.85 % (32329)------------------------------
% 0.30/1.85 % (32329)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 0.30/1.85 % (32329)Termination reason: Unknown
% 0.30/1.85 % (32329)Termination phase: shuffling
% 0.30/1.85
% 0.30/1.85 % (32329)Memory used [KB]: 1279
% 0.30/1.85 % (32329)Time elapsed: 0.002 s
% 0.30/1.85 % (32329)Instructions burned: 3 (million)
% 0.30/1.85 % (32329)------------------------------
% 0.30/1.85 % (32329)------------------------------
% 0.30/1.85 % (32335)dis+21_1:1_cbe=off:cnfonf=off:fs=off:fsr=off:hud=1:inj=on:i=3:si=on:rtra=on_0 on DTF2THF_32211 for (2999ds/3Mi)
% 0.30/1.85 % (32332)Instruction limit reached!
% 0.30/1.85 % (32332)------------------------------
% 0.30/1.85 % (32332)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 0.30/1.85 % (32332)Termination reason: Unknown
% 0.30/1.85 % (32332)Termination phase: Property scanning
% 0.30/1.85
% 0.30/1.85 % (32332)Memory used [KB]: 1535
% 0.30/1.85 % (32332)Time elapsed: 0.006 s
% 0.30/1.85 % (32332)Instructions burned: 15 (million)
% 0.30/1.85 % (32332)------------------------------
% 0.30/1.85 % (32332)------------------------------
% 0.30/1.85 % (32335)Instruction limit reached!
% 0.30/1.85 % (32335)------------------------------
% 0.30/1.85 % (32335)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 0.30/1.85 % (32335)Termination reason: Unknown
% 0.30/1.85 % (32335)Termination phase: shuffling
% 0.30/1.85
% 0.30/1.85 % (32335)Memory used [KB]: 1279
% 0.30/1.85 % (32335)Time elapsed: 0.002 s
% 0.30/1.85 % (32335)Instructions burned: 3 (million)
% 0.30/1.85 % (32335)------------------------------
% 0.30/1.85 % (32335)------------------------------
% 0.30/1.85 % (32337)lrs+1002_1:1_aac=none:au=on:cnfonf=lazy_gen:plsq=on:plsqc=1:plsqr=4203469,65536:i=1041:si=on:rtra=on_0 on DTF2THF_32211 for (2999ds/1041Mi)
% 0.30/1.86 % (32330)Instruction limit reached!
% 0.30/1.86 % (32330)------------------------------
% 0.30/1.86 % (32330)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 0.30/1.86 % (32330)Termination reason: Unknown
% 0.30/1.86 % (32330)Termination phase: Property scanning
% 0.30/1.86
% 0.30/1.86 % (32330)Memory used [KB]: 1791
% 0.30/1.86 % (32330)Time elapsed: 0.011 s
% 0.30/1.86 % (32330)Instructions burned: 38 (million)
% 0.30/1.86 % (32330)------------------------------
% 0.30/1.86 % (32330)------------------------------
% 0.30/1.86 % (32339)lrs+10_1:1_av=off:chr=on:plsq=on:slsq=on:i=7:si=on:rtra=on_0 on DTF2THF_32211 for (2999ds/7Mi)
% 0.30/1.86 % (32339)Instruction limit reached!
% 0.30/1.86 % (32339)------------------------------
% 0.30/1.86 % (32339)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 0.30/1.86 % (32339)Termination reason: Unknown
% 0.30/1.86 % (32339)Termination phase: shuffling
% 0.30/1.86
% 0.30/1.86 % (32339)Memory used [KB]: 1407
% 0.30/1.86 % (32339)Time elapsed: 0.004 s
% 0.30/1.86 % (32339)Instructions burned: 9 (million)
% 0.30/1.86 % (32339)------------------------------
% 0.30/1.86 % (32339)------------------------------
% 0.30/1.86 % (32324)First to succeed.
% 0.30/1.86 % (32342)lrs+10_1:1_acc=on:amm=sco:cs=on:tgt=full:i=16:si=on:rtra=on_0 on DTF2THF_32211 for (2999ds/16Mi)
% 0.30/1.86 % (32343)lrs+21_1:1_au=on:cnfonf=off:fd=preordered:fe=off:fsr=off:hud=11:inj=on:kws=precedence:s2pl=no:sp=weighted_frequency:tgt=full:i=3:si=on:rtra=on_0 on DTF2THF_32211 for (2999ds/3Mi)
% 0.30/1.86 % (32324)Refutation found. Thanks to Tanya!
% 0.30/1.86 % SZS status Theorem for DTF2THF_32211
% 0.30/1.86 % SZS output start Proof for DTF2THF_32211
% 0.30/1.86 thf(type_def_5, type, nat: $tType).
% 0.30/1.86 thf(type_def_7, type, list_c1059388851t_unit: $tType).
% 0.30/1.86 thf(type_def_8, type, config256849571t_unit: $tType).
% 0.30/1.86 thf(func_def_0, type, list_c1059388851t_unit: $tType).
% 0.30/1.86 thf(func_def_1, type, config256849571t_unit: $tType).
% 0.30/1.86 thf(func_def_2, type, nat: $tType).
% 0.30/1.86 thf(func_def_3, type, minus_minus_nat: nat > nat > nat).
% 0.30/1.86 thf(func_def_4, type, one_one_nat: nat).
% 0.30/1.86 thf(func_def_5, type, plus_plus_nat: nat > nat > nat).
% 0.30/1.86 thf(func_def_6, type, prefix1615116500t_unit: list_c1059388851t_unit > list_c1059388851t_unit > $o).
% 0.30/1.86 thf(func_def_7, type, last_c571238084t_unit: list_c1059388851t_unit > config256849571t_unit).
% 0.30/1.86 thf(func_def_8, type, nil_co1338500125t_unit: list_c1059388851t_unit).
% 0.30/1.86 thf(func_def_9, type, nth_co1649820636t_unit: list_c1059388851t_unit > nat > config256849571t_unit).
% 0.30/1.86 thf(func_def_10, type, size_s1406904903t_unit: list_c1059388851t_unit > nat).
% 0.30/1.86 thf(func_def_11, type, ord_less_nat: nat > nat > $o).
% 0.30/1.86 thf(func_def_12, type, ord_less_eq_nat: nat > nat > $o).
% 0.30/1.86 thf(func_def_13, type, fe: nat > list_c1059388851t_unit).
% 0.30/1.86 thf(func_def_14, type, n0: nat).
% 0.30/1.86 thf(func_def_15, type, n: nat).
% 0.30/1.86 thf(func_def_30, type, sP0: (nat > $o) > (nat > $o) > (nat > $o) > (nat > $o) > $o).
% 0.30/1.86 thf(func_def_31, type, sP1: (nat > $o) > (nat > $o) > (nat > $o) > (nat > $o) > $o).
% 0.30/1.86 thf(func_def_32, type, sP2: (nat > $o) > (nat > $o) > (nat > $o) > (nat > $o) > $o).
% 0.30/1.86 thf(func_def_33, type, sP3: nat > nat > nat > $o).
% 0.30/1.86 thf(func_def_34, type, sP4: nat > nat > nat > $o).
% 0.30/1.86 thf(func_def_35, type, sP5: nat > nat > nat > $o).
% 0.30/1.86 thf(func_def_36, type, sP6: (nat > $o) > (nat > $o) > (nat > $o) > (nat > $o) > $o).
% 0.30/1.86 thf(func_def_37, type, sK7: (nat > nat > $o) > nat).
% 0.30/1.86 thf(func_def_38, type, sK8: (nat > nat > $o) > nat).
% 0.30/1.86 thf(func_def_39, type, sK9: (nat > nat > $o) > nat).
% 0.30/1.86 thf(func_def_40, type, sK10: (nat > nat > $o) > nat).
% 0.30/1.86 thf(func_def_41, type, sK11: (nat > nat > $o) > nat).
% 0.30/1.86 thf(func_def_42, type, sK12: (nat > nat > $o) > nat).
% 0.30/1.86 thf(func_def_43, type, sK13: (nat > nat > $o) > nat).
% 0.30/1.86 thf(func_def_44, type, sK14: (nat > nat > $o) > nat).
% 0.30/1.86 thf(func_def_45, type, sK15: (nat > nat > $o) > nat).
% 0.30/1.86 thf(func_def_46, type, sK16: list_c1059388851t_unit > list_c1059388851t_unit > nat).
% 0.30/1.86 thf(func_def_47, type, sK17: nat > nat > nat).
% 0.30/1.86 thf(func_def_48, type, sK18: (nat > nat) > nat).
% 0.30/1.86 thf(func_def_49, type, sK19: (nat > nat) > nat).
% 0.30/1.86 thf(func_def_50, type, sK20: (nat > nat) > nat).
% 0.30/1.86 thf(func_def_51, type, sK21: (nat > nat) > nat).
% 0.30/1.86 thf(func_def_52, type, sK22: nat > nat).
% 0.30/1.86 thf(func_def_53, type, sK23: nat > nat).
% 0.30/1.86 thf(func_def_54, type, sK24: nat > nat).
% 0.30/1.86 thf(func_def_55, type, sK25: (nat > $o) > (nat > $o) > (nat > $o) > (nat > $o) > nat).
% 0.30/1.86 thf(func_def_56, type, sK26: nat > (nat > $o) > (nat > $o) > nat).
% 0.30/1.86 thf(func_def_57, type, sK27: nat > (nat > $o) > (nat > $o) > nat).
% 0.30/1.86 thf(func_def_58, type, sK28: (nat > nat) > nat).
% 0.30/1.86 thf(func_def_59, type, sK29: (nat > nat) > nat).
% 0.30/1.86 thf(func_def_60, type, sK30: (nat > nat) > nat).
% 0.30/1.86 thf(func_def_61, type, sK31: (nat > nat) > nat).
% 0.30/1.86 thf(func_def_62, type, sK32: (nat > $o) > (nat > $o) > (nat > $o) > (nat > $o) > nat).
% 0.30/1.86 thf(func_def_63, type, sK33: nat > (nat > $o) > (nat > $o) > nat).
% 0.30/1.86 thf(func_def_64, type, sK34: nat > (nat > $o) > (nat > $o) > nat).
% 0.30/1.86 thf(func_def_65, type, sK35: (nat > $o) > (nat > $o) > (nat > $o) > (nat > $o) > nat).
% 0.30/1.86 thf(func_def_66, type, sK36: nat > (nat > $o) > (nat > $o) > nat).
% 0.30/1.86 thf(func_def_67, type, sK37: nat > (nat > $o) > (nat > $o) > nat).
% 0.30/1.86 thf(func_def_68, type, sK38: nat > list_c1059388851t_unit).
% 0.30/1.86 thf(func_def_69, type, sK39: nat > nat).
% 0.30/1.86 thf(func_def_70, type, sK40: (nat > $o) > nat).
% 0.30/1.86 thf(func_def_71, type, sK41: (nat > $o) > nat > nat).
% 0.30/1.86 thf(func_def_72, type, sK42: (nat > $o) > nat > nat).
% 0.30/1.86 thf(func_def_73, type, sK43: nat > (nat > config256849571t_unit > $o) > list_c1059388851t_unit).
% 0.30/1.86 thf(func_def_74, type, sK44: nat > (nat > config256849571t_unit > $o) > nat).
% 0.30/1.86 thf(func_def_75, type, sK45: nat > (nat > config256849571t_unit > $o) > config256849571t_unit).
% 0.30/1.86 thf(func_def_76, type, sK46: list_c1059388851t_unit > nat > (nat > config256849571t_unit > $o) > nat).
% 0.30/1.86 thf(func_def_77, type, sK47: nat > nat).
% 0.30/1.86 thf(func_def_78, type, sK48: (nat > nat) > nat).
% 0.30/1.86 thf(func_def_79, type, sK49: (nat > nat) > nat).
% 0.30/1.86 thf(func_def_80, type, sK50: nat > nat).
% 0.30/1.86 thf(func_def_81, type, sK51: (nat > nat) > nat).
% 0.30/1.86 thf(func_def_82, type, sK52: (nat > nat) > nat).
% 0.30/1.86 thf(func_def_83, type, sK53: nat > nat > nat).
% 0.30/1.86 thf(func_def_84, type, sK54: (list_c1059388851t_unit > $o) > list_c1059388851t_unit).
% 0.30/1.86 thf(func_def_85, type, sK55: (nat > nat) > nat).
% 0.30/1.86 thf(func_def_86, type, sK56: (nat > nat) > nat).
% 0.30/1.86 thf(func_def_87, type, sK57: nat > nat).
% 0.30/1.86 thf(func_def_88, type, sK58: nat > nat).
% 0.30/1.86 thf(func_def_89, type, sK59: (nat > nat) > nat).
% 0.30/1.86 thf(func_def_90, type, sK60: (nat > nat) > nat).
% 0.30/1.86 thf(func_def_91, type, sK61: nat > nat).
% 0.30/1.86 thf(func_def_92, type, sK62: (nat > nat) > nat).
% 0.30/1.86 thf(func_def_93, type, sK63: (nat > nat) > nat).
% 0.30/1.86 thf(func_def_94, type, sK64: nat > nat).
% 0.30/1.86 thf(func_def_95, type, sK65: (nat > $o) > nat).
% 0.30/1.86 thf(func_def_96, type, sK66: (nat > $o) > nat > nat).
% 0.30/1.86 thf(func_def_97, type, sK67: (nat > $o) > nat).
% 0.30/1.86 thf(func_def_98, type, sK68: (nat > nat) > nat).
% 0.30/1.86 thf(func_def_99, type, sK69: (nat > nat) > nat).
% 0.30/1.86 thf(func_def_100, type, sK70: (nat > $o) > nat).
% 0.30/1.86 thf(func_def_101, type, sK71: (nat > $o) > (nat > $o) > (nat > $o) > (nat > $o) > nat).
% 0.30/1.86 thf(func_def_102, type, sK72: nat > (nat > $o) > (nat > $o) > nat).
% 0.30/1.86 thf(func_def_103, type, sK73: nat > (nat > $o) > (nat > $o) > nat).
% 0.30/1.86 thf(func_def_105, type, ph75: !>[X0: $tType]:(X0)).
% 0.30/1.86 thf(f1345,plain,(
% 0.30/1.86 $false),
% 0.30/1.86 inference(subsumption_resolution,[status(thm)],[f1344,f1182])).
% 0.30/1.86 thf(f1182,plain,(
% 0.30/1.86 ((fe @ n) != nil_co1338500125t_unit)),
% 0.30/1.86 inference(cnf_transformation,[status(thm)],[f1])).
% 0.30/1.86 thf(f1,axiom,(
% 0.30/1.86 ((fe @ n) != nil_co1338500125t_unit)),
% 0.30/1.86 file('/export/starexec/sandbox/tmp/tmp.iTgpky4eW3/DTF2THF_32211.p',fact_0_FeNNotEmpty)).
% 0.30/1.86 thf(f1344,plain,(
% 0.30/1.86 ((fe @ n) = nil_co1338500125t_unit)),
% 0.30/1.86 inference(trivial_inequality_removal,[status(thm)],[f1342])).
% 0.30/1.86 thf(f1342,plain,(
% 0.30/1.86 ((fe @ n) = nil_co1338500125t_unit) | ((last_c571238084t_unit @ (fe @ n)) != (last_c571238084t_unit @ (fe @ n)))),
% 0.30/1.86 inference(superposition,[status(thm)],[f1001,f1029])).
% 0.30/1.86 thf(f1029,plain,(
% 0.30/1.86 ( ! [X0 : list_c1059388851t_unit] : (((last_c571238084t_unit @ X0) = (nth_co1649820636t_unit @ X0 @ (minus_minus_nat @ (size_s1406904903t_unit @ X0) @ one_one_nat))) | (nil_co1338500125t_unit = X0)) )),
% 0.30/1.86 inference(cnf_transformation,[status(thm)],[f655])).
% 0.30/1.86 thf(f655,plain,(
% 0.30/1.86 ! [X0 : list_c1059388851t_unit] : ((nil_co1338500125t_unit = X0) | ((last_c571238084t_unit @ X0) = (nth_co1649820636t_unit @ X0 @ (minus_minus_nat @ (size_s1406904903t_unit @ X0) @ one_one_nat))))),
% 0.30/1.86 inference(ennf_transformation,[status(thm)],[f2])).
% 0.30/1.86 thf(f2,axiom,(
% 0.30/1.86 ! [X0 : list_c1059388851t_unit] : ((nil_co1338500125t_unit != X0) => ((last_c571238084t_unit @ X0) = (nth_co1649820636t_unit @ X0 @ (minus_minus_nat @ (size_s1406904903t_unit @ X0) @ one_one_nat))))),
% 0.30/1.86 file('/export/starexec/sandbox/tmp/tmp.iTgpky4eW3/DTF2THF_32211.p',fact_1_LastIsLastIndex)).
% 0.30/1.86 thf(f1001,plain,(
% 0.30/1.86 ((last_c571238084t_unit @ (fe @ n)) != (nth_co1649820636t_unit @ (fe @ n) @ (minus_minus_nat @ (size_s1406904903t_unit @ (fe @ n)) @ one_one_nat)))),
% 0.30/1.86 inference(cnf_transformation,[status(thm)],[f574])).
% 0.30/1.86 thf(f574,plain,(
% 0.30/1.86 ((last_c571238084t_unit @ (fe @ n)) != (nth_co1649820636t_unit @ (fe @ n) @ (minus_minus_nat @ (size_s1406904903t_unit @ (fe @ n)) @ one_one_nat)))),
% 0.30/1.86 inference(flattening,[status(thm)],[f216])).
% 0.30/1.86 thf(f216,negated_conjecture,(
% 0.30/1.86 ~((last_c571238084t_unit @ (fe @ n)) = (nth_co1649820636t_unit @ (fe @ n) @ (minus_minus_nat @ (size_s1406904903t_unit @ (fe @ n)) @ one_one_nat)))),
% 0.30/1.86 inference(negated_conjecture,[status(cth)],[f215])).
% 0.30/1.86 thf(f215,conjecture,(
% 0.30/1.86 ((last_c571238084t_unit @ (fe @ n)) = (nth_co1649820636t_unit @ (fe @ n) @ (minus_minus_nat @ (size_s1406904903t_unit @ (fe @ n)) @ one_one_nat)))),
% 0.30/1.86 file('/export/starexec/sandbox/tmp/tmp.iTgpky4eW3/DTF2THF_32211.p',conj_0)).
% 0.30/1.86 % SZS output end Proof for DTF2THF_32211
% 0.30/1.86 % (32324)------------------------------
% 0.30/1.86 % (32324)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 0.30/1.86 % (32324)Termination reason: Refutation
% 0.30/1.86
% 0.30/1.86 % (32324)Memory used [KB]: 6652
% 0.30/1.86 % (32324)Time elapsed: 0.026 s
% 0.30/1.86 % (32324)Instructions burned: 81 (million)
% 0.30/1.86 % (32324)------------------------------
% 0.30/1.86 % (32324)------------------------------
% 0.30/1.86 % (32315)Success in time 0.035 s
%------------------------------------------------------------------------------